Yes! I wanted to mention him but I would have no idea which song to post. They have all been bona-fide hilarious and extremely well done!
I know, and I just realized I haven’t seen the interview with Mike Johnson one! I sent it to myself so I can watch when I get home from work. 
By the bye, how are you guys posting video clips? I don’t seem to have permission.
This doesn’t fit the “famous song rewritten with comedic lyrics” definition of parody, although the title is clearly a parody of the hymn “Jesus Loves Me This I Know”. The Austin Lounge Lizards – “Jesus Loves Me But He Can’t Stand You”
(And apparently Discourse can’t stand the song, because it won’t let me embed the video no matter what I do)
How about “Beat It”, the parody Michael Jackson wrote of Weird Al’s completely original song “Eat It”?
Do I have to spell it out?
Speaking of Weird Al, my wife just came into the living room, all excited about some shoes she’d just bought. So it was hard to resist launching into:
Some girls like to buy new shoes
and others like driving trucks and wearing tattoos…
How about apropos but amusing lyrics to classical pieces? Like:
Please keep your bull outside the china shop
No bulls allowed, that’s where they stop
to the tune of “March of the Toreadors” from Carmen
Please don’t play your violin at night
Wolfgang, go to bed, turn out the light
To Eine Kleine Nachtmusik.
Or turning “In the Hall of the Mountain King” into a countdown:
Here’s a bit more of the same
there’s just one man to blame
Grieg is the composer’s name,
eight more to go
I had that running through my head during my last few weeks before retirement!
All courtesy of Beethoven’s Wig.
Richard Cheese does great lounge act versions of hit songs, totally subverting them. Though he rarely changes the words, just the style. And a perfect album name: Lounge Against the Machine.
And there’s a series on YouTube called “Regurge”, they do Mime After Mime, a parody of Cyndi Lauper.
This entire thread is both of those and more. Let’s not try to disqualify people’s wonderfully humorous songs by nitpicking.
And this has been stuck in my head for years and years now…
Ivor Biggun managed to parody the entirety of the 80s with Bras on 45.
If we’re going this route, I highly suggest Metalachi. They play hard rock and metal classics in a mariachi style. It’s surprisingly good, and they put on a great show as well!
Diner Lobster (SNL spoof of “Who Am I?” from Les Miserables):
I’m surprised nobody mentioned Capitol Steps yet; they had some good ones. My personal favorites are Who’ll Drop a Bomb on Saddam Saddam Saddam (couldn’t find a link) and Bomb Iraq.
My #1 favorite of all time, though? No contest. 
Two of my favorites are by Tom Lehrer:
National Brotherhood Week
And
Fugue for Scientists
I can’t find a version on youtube, you will have to sing it yourself. But here are the lyrics
Also, Tom Lehrer has put all his work in the public domain.
I haven’t read the entire thread, but the old Styx “classic” known as “The Plexiglass Toilet Choir” was a self-parody. People generally do not believe it exists until they hear it, and sometimes not even then.
It was sung by John Curulewski, their original second guitarist who left the band after 1975’s “Equinox”, which had “Lorelei” on it; he was replaced by Tommy Shaw and died from natural causes in the late 1980s. He was about a foot taller than his bandmates, and I wonder if he may have had Marfan syndrome.
There’s a better Baby Got Back version. In fact it was so good that Glee swiped the arrangement without attribution or payment. Jerks.
In a similar vein, 99 Problems is also greatly improved by a little banjo.
In both cases, it’s the same words as the original with new music applied.
In the category of “funny versions of classical music” (and setting aside the excellent work of Peter Schickele for a moment), you have to go with Spike Jones Is Murdering The Classics!.
Finally, here’s an overproduced Rockapella having a bit of fun with the Bach “Little” G Minor Fugue and Beethoven’s Ode to Joy. “Women clearly are much smarter than we ever thought they’d be.”
It’s only part of the song but Adrian Belew does a great Bob Dylan impression on Flakes by Frank Zappa.
I got the tape of The Serpent is Rising from someone. I thought that maybe they had recorded the Plexiglass Toilet song over a real song.
Is On Top of Spaghetti a parody?
That’s another Beethoven’s Wig piece.
ETA: The trucker in the animation bears a close resemblance to Richard Perlmutter of Beethoven’s Wig, but it looks like that’s the closest they come to giving an attribution.