What do you call a cover that changes the lyrics of the song, for example for a humorous effect (comedians or American late show hosts occasionally do that)? Is there a special term for that?
Parody music, or musical parody, involves changing or copying existing (usually well known) musical ideas, and/or lyrics, or copying the particular style of a composer or performer, or even a general style of music. Although the intention of a musical parody may be humour (as in burlesque), it is the re-use of music that is the original defining feature.
In music, parody has been used for many different purposes and in various musical contexts: as a serious compositional technique, as an unsophisticated re-use of well-known melody to present new words, and as an intentionally humorous, even mocking, reworking of existing musical material, sometimes for satirical effect.