Listen, the reason why there wouldn't be a lawsuit is becuase it's "The 1 Ring" not "The One Ring". Besides...J.R.R isn't around anymore and Christopher wouldn't care.

Actually, no, that's a common misconception.  There are limitations on how much of the original work you can use in a parody.  How much you can use is a grey area, but it generally would not include using an entire melody note for note, as Al does.

ANY musician (even one not doing a parody) can record someone else's song without permission, as long as you pay royalties.  This is called a "compulsory license."  You are of course free to negotiate with the copyright holder for a lower rate.

Weird Al talks about negotiating royalties on his web site here:

http://www.weirdal.com/aaarchive.htm

so apparently he does pay them.