If you continue reading the case you cite, you’ll see they go on to discuss how “parody” does not have to be as direct as satire. And they accept that a 2 live crew song sampling Roy Orbison’s “Pretty Woman” can be parody simply because its shocking nature comments upon the bland nature of the Orbison song. Weird Al would have pretty similar arguments.
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Idk a lawyer is going to be able to argue that any of Weird Al’s songs “at least in part” comment on the original work. I think it’s fair to say that getting permission avoids legal risk, but not that his songs are definitely not covered as a parody.