I was wondering if there was a name for when different instruments play the same phrase in succession without overlapping, like in the outro of Moose the Mooche by Charlie Parker.
This is called 'imitation'.
Imitation is the repetition of a melody in a polyphonic texture shortly after its first appearance in a different voice. The melody may vary through transposition, inversion, or otherwise, but retain its original character. The intervals and rhythms of an imitation may be exact or modified. [Wikipedia]
In a Jazz context a repeated phrase (or 'ostinato') is called a 'riff', although in 'Moose the Mooche' the repeated phrase is a melodic fragment rather than a riff.
Call and response seems to me the closest thing to what you describe.
If you google "call and response" you'll find several articles on the subject, many of which imply that the "call" part and the "response" part are not the same, and that's true in many cases, but the term definitely also applies to situations in which the call and the response are the same (same phrase) but played by different instruments or different singers. (e.g. one individual sings the "call" phrase, and a group of other singers repeats it together, something which is common in religious/spiritual musical contexts)
A couple of articles on the subject:
(And many more can be found with a search for "call and response")
Variation would be a good word for that, even though it is a bit more general. Quoting Wikipedia:
In music, variation is a formal technique where material is repeated in an altered form. The changes may involve melody, rhythm, harmony, counterpoint, timbre, orchestration or any combination of these.
The same phrase played by different instruments successively would be a simple orchestration variation.
Another term I've seen used to describe this is "antiphony" or "polychoral antiphony", which is often used to describe this call and response in choir music.
The looser term antiphony is generally used for any call and response style of singing, such as the kirtan or the sea shanty and other work songs, and songs and worship in African and African-American culture. Antiphonal music is that performed by two choirs in interaction, often singing alternate musical phrases. Antiphonal psalmody is the singing or musical playing of psalms by alternating groups of performers. The term “antiphony” can also refer to a choir-book containing antiphons.
When two or more groups of singers sing in alternation, the style of music can also be called polychoral [antiphony]. Specifically, this term is usually applied to music of the late Renaissance and early Baroque periods.
Same wikipedia article