I'm looking specifically for the principles of improvisation as demonstrated by Charlie Parker and Thelonious Monk, the fathers of bebop. I realize my opinion may be controversial but I see Dizzy Gillespie as the foremost promoter of bebop, not one of the inventors, a view I share with Monk.
A really really important principle that bebop players used was that they improvised on top of the melody and not the harmony.
Another important thing was they didn't think of the harmony as a whole, but they thought out every single chord change.
Let's take Parker's Moose the Mooch for instance:
When Charlie Parker improvised on top of those chord changes, he didn't think "I'm in Bb major, so I can solo on top of that". At the first bar he thought the notes of the chord Bb, as well as the notes of the melody. On the first two beats of the second bar, he thought the notes of the chord C- as well as the notes of the melody etc.
Charlie Parker's improvisational technique was largely based on formulas (or riffs) that he would transpose to fit chord progressions. He would have to do this in order to play as fast and as fluently as he did, even when he was high and this is in no way a criticism of his creativity--he did invent an entire musical language using his riffs as words. A statistical analysis of Parker's formulas can be found in Toward a cognitive analysis of creativity: Improvisation in jazz by Weisberg et. al. and a musical analysis by close reading of the Charlie Parker Omnibook.
Thelonious Monk's contribution was along multiple reinforcing lines. First there were his reharmonizations, which included chords, tone clusters and even microtones--his use of minor seconds was his way of trying to play the note in the crack between the keys. There was his strong rhythm combined with an amazing use of space. And then there was his innovative use of the blues scale to create angular melodies with repeating melodic fragments. To hear just how different his musical sensibilities were, listen to his version of "Tea for Two."
A detailed analysis of Monk's improvisation style can be found in The Musical Language of Thelonious Monk by Eddie S Meadows
Some characteristics of Monk's playing are his use of fragments of the song's theme. Often he would repeat a phrase or a part of a phrase, as if breaking it down into smaller parts, interrupting it with some of his other signature figures such as a whole tone descending scale.
I'm less familiar with Charlie Parker, but maybe you could find out something interesting by having a look at the harmonic content of the solo and how it relates to the underlying chords. You might also try to analyse the melodic profile [*] once you have the solos transcribed and see if that leads to any insights.
[*] For some ideas, see Friedmann (1985): A methodology for the discussion of contour, Journal of Music Theory, 29:2.