Actually, he speeds the music up a bit, because he thinks it is funnier, and occasionally removes or adds verses to make it work for his idea, but for no other 'legal' reasons.This is......... inaccurate.
Parody does not give one free reign to use the underlying music of a song whose lyrics are being completely rewritten. In fact, the 'transformative' test can't credibly be applied to Weird Al song expressly because he's not transforming the nature of the music. In other words, it still has to sound pretty much exactly like "Smells Like Teen Spirit" or "Beat It" for the joke lyrics to work. Thus, that aspect of the copyright is intact, and permission must be sought from the respective copyright owner (who is not often the artist in question). There is also the blatantly commercial nature of the new work, which puts a thumb on the scale in weighing fair use. (As expressly set forth in the statute)
He does also ask the artist's permission out of courtesy, but legally he still has to clear it with the copyright owner. (The copyright owner might be BMG, or Harry Fox or whoever; if they say yes but the artist says no, Al won't make the song.)
While 'Weird Al' asks permission, and will not put any song that is not approved on an album, he is not legally obligated to do so. He is far from the only artist who creates parodies, almost none of them ask, and most of them make a profit off of their songs. One of the most prolific is Bob Rivers, who started doing stand up, and ended up working at a radio station, where he could even hire people who sound like the original artists to perform the tracks.
As long as you are changing a majority of the lyrics, so that no one can reasonably think it is the same song, there is really no legal issue. You are not required to change everything, only enough to make it 'different'.
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As it relates to the game though, Trumps, Clinton, and Il, they are parodies, even changing the names of the Trumps, and they are drawn as caricatures, whereas Titania was an homage.