He doesn't actually make very heavy use of the satire plank of fair use. He credits the original artists. From his own website
"Does Al get permission to do his parodies?
Al does get permission from the original writers of the songs that he parodies. While the law supports his ability to parody without permission, he feels it’s important to maintain the relationships that he’s built with artists and writers over the years. Plus, Al wants to make sure that he gets his songwriter credit (as writer of new lyrics) as well as his rightful share of the royalties."
The fact that he could rely on fair use is separate from whether he as an artist does rely on fair use.
"Does Al get permission to do his parodies?
Al does get permission from the original writers of the songs that he parodies. While the law supports his ability to parody without permission, he feels it’s important to maintain the relationships that he’s built with artists and writers over the years. Plus, Al wants to make sure that he gets his songwriter credit (as writer of new lyrics) as well as his rightful share of the royalties."
The fact that he could rely on fair use is separate from whether he as an artist does rely on fair use.