Bob Rivers has done several holiday tunes in the vein of the OP.
The Most Fattening Time of the Year (The Most Wonderful Time of the Year)
There’s a Santa Who Looks a Lot Like Elvis (It’s Beginning to Look a Lot Like Christmas)
There’s Another Santa Claus (Here Comes Santa Claus)
Police Stop My Car (Felix Navidad)
Chipmunks Roasting on an Open Fire (Chestnuts Roasting on an Open Fire)
More Bob Rivers:
Walkin 'Round in Women’s Underwear (Winter Wonderland)
The Twelve Pains of Christmas
Wreck the Malls
O Come All Ye Grateful Deadheads
The Restroom Door Said “Gentlemen”
Now I gotta go dig out that CD! Thanks for the reminder. 
We listen every year now to “Because I’m Santa”, which is Kelly Ogden of The Dollyrots singing alternate lyrics to their signature song “Because I’m Awesome” (like in that Kohl’s commercial).
I left one out. “The Rummy Rocker Boy.” It’s a spoof of the Bing Crosby/David Bowie duet of “The Little Drummer Boy” with a Bing Crosby and Ozzy Osbourne soundalike.
I expect everyone’s seen this by now, but I still get a kick out of the Star Trek TNG version of “Let It Snow.”
Jingle Bells
https://www.youtube.com/watch?v=uztkwpwPo_U
Walkin’ in an Essex Wonderland
http://98online.com/media/essex-wonderland/
Holy Shit It’s Christmas - not really a parody but one of my favorites
https://www.youtube.com/watch?v=ZR3EiIO22GQ
Beat me to it, so I will go for another favorite: The 12 days of Christmas, Indian style. Performed by Boymongoose.
♬ Once a Good King went to sea
And his name was Wencil
He ate oysters so that he
Could put lead in his pencil
Buggering the cabin boy, Wencil thought was dandy
Wencil was a goodly King, but a trifle randy ♬
(Yeah, I made that one up myself.)
The US Navy Band does it better.
Peter Schickele brought us three by P.D.Q. Bach:
Throw The Yule Log On, Uncle John
O Little Town of Hackensack
Good King Kong
Wow. I’d never heard of Grandma Got Run Over By a Reindeer, but judging from the insane number of parodies on youtube, it’s clearly graduated to Christmas Classic.
I’m surprised no one, myself included, has rememered Allan Sherman’s The Twelve Days of Christmas yet.
*On the first day of Christmas, my true love gave to me
A Japanese transistor radio.*
The very first Carols for a Cure CD, issued in 1999, had 12 Days of Phantom.
Still one of the best parody songs, and Carols for a Cure songs, ever.
He’s pretty obscure these days, but I’m surprised that apparently no other Doper has heard of Yogi Yorgesson.
I Yust go Nuts at Christmas
Yingle Bells
Harry Stewart - WikipediaHe’s got a lot of other fun songs, too.