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.
The following tags implicate this tag: multiple_crossover.