(yes, in theory, you can parody the music itself, but again for that to qualify you'll need to write a distinct song. since you're a weird al fan, his "stylistic parodies" where he writes a song in the _style_ of another band without leaning on any existing songs to do it. examples include "dare to be stupid," a stylistic parody of devo which is a legally distinct song, or "mr. popeil," which is a stylistic parody of the B-52s but also not any of their songs.)

regardless, though, in practical terms? unless you go over like 1,000 sales, nobody is going to care.

Al says he gets approval from original rights holders before releasing

He might still play, say Laundry Day for instance, in concert even if it's not on an album

Whether it's legal or not is probably a different question.

Yes, but.

Here is Cornell Law's summary: https://www.law.cornell.edu/wex/parody