Weird Al doesn't do covers, aside from a few exceptions. He does parodies. Parody falls under fair use, so legally I don't think he has to pay royalties, though he may anyway.
I'm not a lawyer, and most of my knowledge of copyright law comes from Tom Scott's video on the subject [1], but it's worth noting that it's not universally agreed on that Weird Al's parodies fall under fair use. Again, I am not a lawyer, at best I'm a novice when it comes to copyright law, but what I do know, I agree with what Tom says: I wouldn't expect Weird Al to win if he was taken to court (if he didn't ask permission).
IIRC Weird Al treats his parodies more like covers (even going so far as to get permission), and there are credits for the composers, etc. So I wouldn't be surprised if they pay out more like covers than parodies.
He likely doesn't want to be in front of the Supreme Court someday having to defend the parodic aspect like the famous Barbie Girl case, especially as it is not clear that all of his songs are parodies of the song and not some other aspect in the format of the song.
Not disagreeing, but you have to look beyond US copyright law. Fair use is less of a thing i.e. in Europe, and his streams clearly come from all over the world. I would just assume his label made deals with the respective copyright holders covering several jurisdictions and markets just to be on the safe side. That doesn't necessarily mean they had to do those deals.
