The thing I liked about the Rutles, and I bought their first album right after “All You Need is Cash” aired, was that the humor was subtle and the songs were well written and performed. I don’t know anyone who lives under a rock to test this, but I suspect someone who never heard of the Beatles could enjoy many of their songs just as songs, even if they didn’t know they were humorous.
I was going to mention Spike Jones. My favorite is “Chloe,” which has one of my favorite jokes in it.
[SPOILER]phone rings
SINGER: Hello? You don’t say! You don’t say! You don’t say!
hangs up
AUDIENCE: Who was it?
SINGER: He didn’t say!
Later in the song
phone rings
SINGER: Hello? You don’t say! You don’t say! You don’t say!
hangs up
AUDIENCE: Who was it?
SINGER: Same guy![/SPOILER]
I agree. They did some good tunes, and the fact that they happened to be parodies was gravy!
As a side note, any fan of The Rutles needs to check out the Utopia album Deface the Music, which is a similar collection of Beatles pastiches, though maybe not quite as humorous.
Yes, that’s correct, Bob Rivers recorded “What if God Smoked Cannabis,” not Weird Al. And it wasn’t Joan Osborne singing, it was a sound-alike singer. That’s one of the things I enjoy about Bob Rivers, he employs singers who sound remarkably like the original artists. But Osborne liked the parody so much, she has performed it in concerts.
mbh already brought up the great Spike Jones, (not to be confused with Spike Jonez.) Between Spike and Weird Al was Allan Sherman, who made parody hay out of not just folk songs (Sarah Jackman from Frere Jacques) and jazz standards (Eight Foot, Two, Solid Blue from Five Foot, Two, Automation from Imagination and the incomparable I See Bones from C’e Ci Bon) to classical melodies (Hello, Muddah, Hello, Fadduh from Dance of the Hours and Hungarian Goulash #5 from Hungarian Rhapsody #5)
Allan Sherman’s entire albums may be heard on Youtube.com .
Please forgive my French spelling. I’m not hip to that.
Sherman apparently wrote a symphonic piece, which I can’t locate. Working with a classic orchestra, he found the trombonists did not have plunger mutes. He went to a hardware store to buy some plumber’s plungers for the trombones. He didn’t bother with the wooden handles, telling the clerk he wouldn’t need them. “Trust me, you’re gonna need the handles,” said the clerk.
The best satirical singer/songwriter of the last half of the twentieth century is Tom Lehrer. He’s better than Allan Sherman, better than Spike Jones, better than Stan Freberg, better than Peter Schickele (P. D. Q. Bach), better than Weird Al Yankovic. Some of his songs were based on previous specific songs, but some were just imitating the general form of certain genres. And he did his singing and songwriting in his spare time, never considering himself a full-time musician.
Corky and the Juice Pigs have a few songs that are more “style” parodies than parodies of specific songs. “REMember” is a parody of R.E.M., and “Janitor” is a parody of Neil Young.
Seriously. Smells Like Teen Spirit has a catchy beat, but it’s literally a bunch of random nonsense. Smells Like Nirvana is hilarious. “It’s hard to bargle nargle nous with all these marbles in my mouth.” 
Cheech and Chong’s “Born In East L.A.” is better than “Born in the U.S.A.”