I got this off Weird Al’s Wikipedia page:
On numerous occasions, Prince has refused Yankovic permission to record parodies of his songs. Yankovic has stated in interviews that he has “approached him every few years [to] see if he’s lightened up.” Yankovic related one story where, before the American Music Awards where he and Prince were assigned to sit in the same row, he got a telegram from Prince’s lawyers, demanding he not make eye contact with the artist.
Paul McCartney, also a Yankovic fan, refused Yankovic permission to record a parody of Wings’ “Live and Let Die”, titled “Chicken Pot Pie”, because McCartney is a vegetarian and found the parody to be improper.
I have new respect for Al. Technically, as long as he pays for the privilege, he can cover/parody anyone he wants to. He asks to maintain good karma in the community. Most artists go along, some even playing on the sessions or appearing in the videos.