I will be using the Eb version of the Omnibook as a reference. I am not reproducing the transcription here, due to copyright considerations. If you are a sax player, you should own the "Omnibook" anyway.
The recording is posted on Youtube here. The pitch seems a bit high.
Following is my best effort at figuring out Bird's musical thinking on this solo.
Head
"Anthropology" is a melody in AABA form, written over "Rhythm changes." Parker used this harmonic framework for quite a few of his tunes. For more about this chord progression, see this article.
Authorship is usually credited jointly to both Parker and Gillespie, although Carl Woideck cites a 1949 article that paraphrases Parker as saying that he (Parker) was the sole composer. Certainly there are Parkerisms throughout this melody: the "Moose the Mooche" rhythm (noted by Koch) in measure 2, the turnaround pattern in mm7-8, the "Ornithology" lick in m17, another "Ornithology" lick in mm23-24. If Dizzy wrote any of it, it would be interesting to know which parts.
A few more comments about the head:
m1: Although the Rhythm changes progression is often played with an Em7 (VI) or E7 (V of II) on beats 3 and 4, the melody does not imply this.
m3, beats 3-4: Em7 is implied here, not E7 as the Omnibook has it.
m6, beats 3-4: Cm is implied here for beats 3 and 4 by the Eb note - a common way to play Rhythm changes.
Solo Chorus 1 (Omnibook lines 8-15)
m1: This scalewise run occurs twice more, later in the solo. Bird used it in the 1945 recording of "Thrivin' On a Riff," which is just "Anthropology" with another title. He is quoting himself.
mm3-4: Parker seems to be outlining Gmaj7 in m3 with a characteristic pickup/triplet figure, contrary to the chords shown in the Omnibook. Beat 4 begins an Am7 arpeggio (starting early on an upcoming chord, or "playing into" the change, is a characteristic Parker device).
m5: Parker likes to use an F natural note in this spot to define the G7 (V of IV). Here he carries the chord forward into the first two beats of m6; this scalewise shape, with the Gb passing tone, is an example of what David Baker would later dub the "bebop scale."
m6, beats 3-4: The outlined Am7 is a "substitution by third" for Cmaj or Cmaj6.
m7: A Cm arpeggio is superimposed here, that would normally occur in the previous measure. But perhaps Parker is hearing it also as Am7b5, preparing an implied D7 in beats 3-4, resolving to G in m8.
mm9-11: A Gmaj arpeggio in m9 is changed into G+ in m10, perhaps reflecting the shape played in m7. Measure 11 is an obvious II V (Am7 D7). This is another Parkerism: outlining a progression that makes sense in itself, but that does not coincide with the chords that are being played by the rhythm section.
m14: F natural note, delayed from its usual placement in m13, indicating a temporary key of C.
m15: Suggestion of D7 in beats 3-4.
(bridge) mm17-24: A beautifully conceived and executed harmonic game, perhaps mapped out to some extent in advance. The standard "Rhythm" bridge follows the circle of fourths (B7 E7 A7 D7), each 2-measure dominant chord resolving into the next. In m17 Parker uses the notes of an E major scale (B7 is V in the key of E) for one measure (think B mixolydian if you like), then outlines an Am7b5 chord over the second bar of the B7. The notes in the Am7b5, played against the B7, come across as b7 (A), b9 (C), 3 (Eb=D#), and #5 or b13 (G). Adding these tensions serves to propel the B7 more strongly into the resolution to E7 that follows. It's not too different from the way Bird often treats bar 4 of a blues (see this article).
In a general sense, this "game" consists of building a m7b5 chord from the seventh of a dominant chord, resulting in the tension notes mentioned above. You might equally well look at the m7b5 shape as the 3-5-7-9 of a tritone substitution - in the case above, Am7b5 is F9 (sub V for B7) minus the root note F. Parker may have been thinking of it either way, or both ways.
In a general sense, this "game" consists of building a m7b5 chord from the seventh of a dominant chord, resulting in the tension notes mentioned above. You might equally well look at the m7b5 shape as the 3-5-7-9 of a tritone substitution - in the case above, Am7b5 is F9 (sub V for B7) minus the root note F. Parker may have been thinking of it either way, or both ways.
mm19-20: He repeats this game with pickups into m19, through beats 1-3, using diatonic A major notes over the E7, followed by an arpeggiated Dm7b5 on beat 4, extended into m20. The pickups into m19 and into m20 again reflect his common practice of "playing into" an upcoming chord change.
mm21-22: Parker plays 6 beats of diatonic D major notes over the A7; in m22, beats 3-4, he plays a shape very close to Gm7b5 - the same game, adding tensions right before the resolution.
mm23-24: This time he uses a different approach, suggesting a II V (Am7 D7). Note the embedded chromatic line A-G#-G-F# (see the Billie's Bounce post for more examples of this type of II V).
m27: Quotes "Blue Champagne."
m27: Quotes "Blue Champagne."
mm28-29: Parker repeats the lick that began his first solo chorus, and plays the three "Blue Champagne" notes again.
Solo Chorus 2 (Omnibook lines 16-23)
mm1-4: Quotes the song "Honey."
m5: "Bebop scale," leading to the F natural note on beat 4.
mm8-10: A pattern probably lifted from an exercise book, that embellishes a G triad with approach notes (scale tones from above, chromatic neighbors from below). This is a spot where Parker seems to have simply thought in G major, without strict reference to the chords. The Omnibook rhythm is not quite right - Bird starts the pattern with three pickup notes, then slides into the notated rhythm.
mm11-12: In contrast to the "just play in G" approach, here Parker plays a turnaround pattern that outlines the chords meticulously.
m13: Characteristic pickup/triplet figure outlining Gmaj7 from third to ninth. As the scale descends in beats 3-4 he hits the F natural that defines the chord as G7.
m14: The note Eb defines a Cm chord in its normal position.
(bridge) mm17-19: Quotes the song "Tenderly" (Omnibook rhythmic interpretation doesn't quite convey this).
mm20-21: Parker plays a scale/fall, landing on a low B. The Omnibook shows only some cryptic noteheads here - perhaps this was a preliminary sketch by the transcriber that the copyist mistakenly inked into the final copy.
mm21-22: Pickup/triplet figure, followed by a Parker signature double-time lick beginning on the alto's high D (see the Omnibook transcriptions of solos on "Now's the Time" and "Billie's Bounce"). The lick is not played in double time here, just in very fast swing beat. It outlines Em7 A7.
m25: The same lick that began this solo in chorus 1, m1.
m29: F natural over the G7.
m30: The Omnibook suggests a C#dim chord here. Parker may or may not be suggesting C#dim; the E natural suggests that in any case he is not suggesting Cm, as he had previously done in the head and solo choruses. Either Cm6 or C#dim7 would be standard in this part of the progression.
Solo chorus 3 (Omnibook lines 24-32)
mm1-4: Quotes the clarinet solo from the dixieland tune "High Society." As in chorus 2, mm17-18, he seems to be reducing his harmonic concept to "just play in G major," although the notes do fall nicely on the chords.
m6: Plays the F natural that more often comes in m5.
m8: A nice II V suggesting Am7 D+7.
mm9-10: In m9 Bird plays a scale-tone thirds pattern (D-B-C-A) in G major that is embellished with chromatic passing tones, then in m10 moves the phrase up a half step, to Ab major - a mildly "outside" sound.
m13: F natural over G7.
m14: The Omnibook authors suggest a Gmaj7 chord here, perhaps thinking of of Parker's characteristic major seventh to major ninth lick. However, I could just as easily hear these notes as suggesting D7, which one might expect here.
I should note that the Omnibook seems inconsistent in the the criteria used to decide which chord changes to print. Sometimes the choice seems to be made on the basis of the notes that Parker is employing; at other times the chords may reflect what the rhythm section is playing, or what the authors think of as "common practice." In addition, there are some disparities in chords between the Eb, Bb, and C editions. It would be good to compare the notated chords with the actual piano chording; unfortunately, in this recording the sound is not clear enough for me to discern exactly what Bud Powell is playing. But going by my impression from other Parker recordings, it seems likely that each person in the band was simply thinking, "It's Rhythm changes," and playing his part with whatever variations he pleased.
(bridge) m18, beats 3-4: Same trick as in the bridge of the first solo chorus - adding tension notes in m18 by superimposing an Am7b5 arpeggio on the B7 chord.
I should note that the Omnibook seems inconsistent in the the criteria used to decide which chord changes to print. Sometimes the choice seems to be made on the basis of the notes that Parker is employing; at other times the chords may reflect what the rhythm section is playing, or what the authors think of as "common practice." In addition, there are some disparities in chords between the Eb, Bb, and C editions. It would be good to compare the notated chords with the actual piano chording; unfortunately, in this recording the sound is not clear enough for me to discern exactly what Bud Powell is playing. But going by my impression from other Parker recordings, it seems likely that each person in the band was simply thinking, "It's Rhythm changes," and playing his part with whatever variations he pleased.
(bridge) m18, beats 3-4: Same trick as in the bridge of the first solo chorus - adding tension notes in m18 by superimposing an Am7b5 arpeggio on the B7 chord.
mm19-20: An A major scale with some chromatic passing tones.
mm 21-22: The signature "double-time" lick, as in solo chorus 2, mm21-22.
mm23-24: 6 beats of suggested Am9, then 2 beats of suggested Ab7 (tritone substitution for D7).
mm25-32: Quotes the song "Temptation." The quote suggests Ab chords (tritone sub for D7) falling nicely over "Rhythm" II V chords.
mm31-32, spilling into the next chorus: A lick similar to one he often used to begin solos on "Cherokee." The notes do not seem to coincide with the chords, suggesting rather something like | D7 G | Am7 D7 | Gmaj9 |. It sounds OK, though.
Some readers may think it unlikely that Parker was really conscious of all these things, especially given the very rapid tempo of this performance. But consider that "Rhythm changes" tunes were a staple of the day (almost always in this key, concert Bb), and that Parker had played and heard many, many thousands of choruses over these changes by the time this performance was recorded. Consider also that he was a truly gifted musician, and was working in an extremely competitive environment. The thought processes that I am suggesting would have been well-practiced for him, to the point of being reflexive.
I hope some readers have found this useful!