Parody implies a character is mimicking a scene, a dialogue, or another series, with the intention of being humorous. A character need not parody another show, they could in fact parody a character or scene from the same series.
Crossover implies that characters from two or more series are joined in one scene.
Some parody pictures are also crossovers, and vice versa, but the two aren't quite equivalent.
Parody might also be a general spoof on given series/character etc. This could be characters acting out of character, a strange scene, and so on, as long as it's intended to be a humorous take different from the usual premise.
source: danbooru wiki