It's a Fair Use defense. I've always read Yankovic's insistence on the artist blessing as a "consent on content"- the actual lyrics that Al's replacing. e.g. Al approaches James Blunt on parodying "You're Beautiful", and Blunt signs off on Al's lyrics. (Atlantic Records eventually revoked permission for "You're Pitiful")
It's a classy thing to do, and I think it keeps the work "above board" and at an Al-standard. I've heard some pop-music parodies that were crass, obnoxious and pretty awful.