Doctor Hook and the Medicine Show:
“Freaking at the Freaker’s Ball”
“The Cover of the Rolling Stone”
“Roland the Roadie and Gertrude the Groupie”
Spike Jones and the City Slickers:
“My Old Flame”
“The William Tell Overture” a.k.a. “Beetlebaum”
“You Always Hurt the One You Love”
“Pal-Yat-Chee”
“Der Fuehrer’s Face”
David Allen Coe “You Never Even Call Me by My Name” a.k.a. “the perfect country-western song”
I’m not sure I could pick a favourite but it would definitely be something by either Flight of the Conchords (maybe Think About It or The Most Beautiful Girl in the Room) or Tenacious D (maybe City Hall or Beelzeboss).
“The Vatican Rag,” by Tom Lehrer; “Lounge Pianist’s Lament” by Lauren Mayers; "Itazuke Tower
You’ve posted some great selections! Been listening to them all morning. Needed a good laugh. Thanks.
Anyone remember Jim Stafford’s Spiders and Snakes? Really funny.
Martin Mull would place high on my list, for Men or Half Hour of Heaven (and Eight Hours of Sleep).
I’ll see that and raise you “The Night Santa Went Crazy”.
With apologies if this one’s been mentioned already, but Weird Al’s “The Saga Begins” is a huge improvement over the actual movie.
If I Didn’t Have You by Tim Minchin
What act is he parodying? (even in his originals he does parody a group or genre)
Well, the OP did say “parody or comic song”. It doesn’t have to be a parody.
for a baseline, Jo Stafford playing it straight
I’m partial to “I’ll hold your hand in mine” by Tom Lehrer - a perfect sad love song! 
Gotta be:
“July 13th, 1985” John Wesley Harding. A song about Live Aid.
“Ten Foot Cock and a Few Hundred Virgins” and “Thank You God” by Tim Minchin
“Chow Down (at Chik-Fil-A)”, “Blurred Bynes” and “Boy is a Bottom” by Willam
“Ghandi Buddha” and its counter version by Cheryl Wheeler
“Three Little Words” by Da Vinci’s Notebook
And echoing previous posters with “Hardware Store” Weird Al Yankovic
Cat’s in the Kettle
For the Canadian Division, I’ll add Bowser and Blue’s “Working Where the Sun Don’t Shine.”
Also, “Carrot Juice is Murder” by the Arrogant Worms.
Forgot about DaVinci’s Notebook. Their send-up of boy-band songs, “Title of the Song,” has got to be one of the best exercises in self-reference ever.
I didn’t know anybody else around here knew that song. Have you heard When the Beatles Hit America?
Excellent choice.
Weird Al Yankovic:
One More Minute
I Lost on Jeopardy
The Saga Begins
King of Suede
Tom Lehrer:
The Vatican Rag
The Elements
Jonathan Coulton:
Baby Got Back
Johnny Cash:
A Boy Named Sue
One Piece at a Time
Wax Audio:
Imagine This (warning: NSFW language at the beginning)