When Charlie Parker was asked to 'sing' a tune, he often didn't. Whether he was trying to sing the melody or not, what he usually did was sing mainly the phrasing, emphasizing the rhythm.
Bebop is a very rhythm-oriented music. In fact, the very name comes from a prominent lick: two eighths followed by a rest … be-BOP.
Now, if we transport our thinking some 7000 miles to the east, we arrive at India, where rhythm is king and written music is a rarity. Their Carnatic music tradition is based on spoken rhythmic patterns passed down from teacher to pupil, from one generation to the next. To master the rhythms, the student must memorize them. It's not sufficient to merely play the rhythms. Knowing -- memorizing -- the verbal patterns is required.
In western music we don't pay much attention to vocalizing rhythms or to memorizing. Maybe we should. Charlie Parker thought in terms of rhythm to the point that he quite readily verbalized a tune's rhythm, often only alluding to its melody. It's a good habit to get into. If we can speak the rhythm of a tune, then we will truly know the rhythmic spirit of the tune.
As drummers, rhythm is pretty much all we have to work with. We're rhythmic animals, so we should be putting all our effort into mastering and interpreting the rhythmic backbone of the tunes we play.
A technique I sometimes use is to transcribe a tune, copying down only the rhythm. Then I have the rhythmic flow in front of me (without all those up and down notes) and I can concentrate on the lines and phrasing. This is especially useful for tunes that use tricky rhythm structures. (It wasn't until I did a rhythmic transcription that I had any idea how to handle Sonny Rollins's “Oleo”) It also helps when the tune’s overall structure is unusual, such as a 44-bar AABA.