Weird Al’s work is often labeled as parody, but not all of it is legally speaking, in addition to the original work he does, Weird Al also does satire. The Supreme Court has addressed the difference between parody and satire. In Campbell v. Acuff-Rose.
Parody needs to mimic an original to make its point, and so has some claim to use the creation of its victim’s (or collective victims’) imagination, whereas satire can stand on its own two feet and so requires justification for the very act of borrowing.
510 U.S. 569, 580-581 (1994). Weird Al’s song, “Perform This Way” which made news a few years ago is a parody. It’s using a work of Lady Gaga’s to comment on Lady Gaga. The song simply wouldn’t have the same effect if Weird Al wrote a song poking fun at Lady Gaga to the tune of a Taylor Swift song.
The same can be said about other great Weird Al works like “Smells like Nirvana,” and to a lesser extent, “White and Nerdy.” But what about “Canadian Idiot” which satirized American’s views of Canadians, or “Eat It,” “Gump,” or “The Saga Begins.”
While the use of “American Idiot,” “Beat It,” “Lump,” and “American Pie” added something to the song, the original songs were not necessary to achieving the goal of the satire, which would weigh against fair use.
The case law surrounding fair use is always in flux, and clear answers are almost never found. But, before publishing a parody of another author’s work it’s good practice to either get permission, like Weird Al, or thing about what you’re commenting on through the use of the work.