While it is true that in jazz, flat keys are generally preferred, I see this as a case where writing it with sharps, especially considering the chords in the turnaround is not a bad choice. Consider:
Amaj7 G#m7 Gmaj7 F#m7
As opposed to:
Bbbmaj7 Abm7 Abbmaj7 Gbm7
As for Dorian and parent keys, the same thing applies to the choice of a flat or sharp key. However, in my experience playing jazz I rarely see songs written in modal key signatures. People prefer to write and read in regular major or minor key signatures and fill in the accidentals. I believe one reason for this is a modal key signature can get confusing if the song has sections that are chromatic. Another is that people think of the tonic note as the central key so say, having a parent key signature of F (or D minor) for A Phrygian is a little odd unless the song is purely in a mode with little to no accidentals.
My last point is that in this case, I donโt see much in the piece to suggest that it is in Dorian mode. The turnaround is very chromatic. The M6 note is used in passing only in the bridge section of the song but it is typical in jazz to use a M6 in tunes that are in a minor key. That doesnโt necessarily make it Dorian, itโs just C# or Db minor in my book.
