I love well-done parodies. One genre that most people don’t know about is Scandinavian-themed Christmas songs.
This one is called “Uncle Sven is Coming to Town.”
This one has been done by various people, but I like this version the best. It’s called “Yingle Bells.” It’s by Harry Stewart, better known as Yogi Yorgesson. His stage persona was that of a Swede who became a Hindu mystic.
I like this one from SNL.
Best “12 Days” ever. It’s the “sheet music” that really sells it, I’ve yet to see any other version incorporate that detail.
Does this count? 
More of a “son parodying how we live Christmas” than a “parody of a carol”.
Who did “Ay yust go nuts at Christmas”?
Harry Stewart as Yogi Yorgesson. https://www.youtube.com/watch?v=1trcQoxJ9qY
It’s a classic but I always like the 12 Pains of Christmas.
My favorite 12 days of Christmas:
Hindu style by Boy Mongoose, Australian band of Indian origin.
Has it been 10 years since it was released? Well, I expect many more years of feeling old… 
Doggie Winter Wonderland
Dog tags ring, are you listening?
In the lane, snow is glistening.
It’s yellow, NOT white, I’ve been there tonight
Marking up my winter wonderland.
Smell that tree? That’s my fragrance.
It’s a sign for wandering vagrants;
"Avoid where I pee, It’s MY property!
Marking up as my winter wonderland.
In the meadow dad will build a snowman
Following the classical design.
Then I’ll lift my leg and let it go, Man,
So all the world will know it’s
MINE - MINE - MINE!!
Straight from me to the fence post
Flows my natural incense dog boast:
“Stay off of my turf, this small piece of earth,
I’ve marked it as my winter wonderland.”
Mad Magazine had a bunch of great Xmas carol parodies in a 1976 issue: Mad Magazine’s 1976 Christmas Issue: Still Relevant, Still Memorized–Years Later – My Media Diary
[quote=“Wesley_Clark, post:2, topic:775531”]
I like this one from SNL.
[/QUOTE]Wonderful song. Darlene Love performing takes it into the top ranks of parodies.
“…See Fiddler on the Roof with actual Jewish actors…”
♬ Once a good king wen 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 ran-an-dy! ♬
.
The Yobs: The 12 Days of Christmas
Tom Lehrer: A Christmas Carol.
A bit dated, since it’s largely about commercialism, which we know is a thing of the past.
Walking 'Round in Women’s Underwear
Bob Rivers was genius at these things.
***MAD ***had great carol parodies every year:
*See the merry Christmas drinking,
fa-la-la-la-la, la-la-la-la!
See the drivers getting stinking,
fa-la-la-la-la, la-la-la-la!
Though their brains are half-corroded,
fa-la-la-la-la, la-la-la-la,
Still they’ll try to drive home loaded!
Fa-la-la-la-la, la-la-la-la!
See the busy intersection,
fa-la-la-la-la, la-la-la-la!
Here come cars from each direction,
fa-la-la-la-la, la-la-la-la!
See the pile-up when they’re meeting,
fa-la-la-la-la, la-la-la-la!
What a novel Christmas greeting!
Fa-la-la-la-la, la-la-la-la!
Hear the sound of sirens blowing,
fa-la-la-la-la, la-la-la-la!
See the Yuletide death toll growing,
fa-la-la-la-la, la-la-la-la!
That’s the fate of drunken drivers,
fa-la-la-la-la, la-la-la-la!
Sing this song to their survivors,
fa-la-la-la-la, la-la-la-la!*
My choir did a funny version of “Good King Kong” by P. D. Q. Bach. For some reason the one posted on Youtube is different from the one I remember.
Oh, dust and smoke, the heater’s broke,
The door just blew away.
I light a match to see the dash
And then I start to pray.
The frame is bent, the muffler went,
The radio, it’s okay.
Oh what fun it is to drive
My Rusty Chevrolet
Only marginally on topic but I heard a folk singer throw out snippits like:
“Check the balls on that big collie”
and
"It’s beginning to look a lot like syphilis "
and from a source I can no longer remember
“Randolph the Bow Legged Cowboy”, a full version with Randolph shooting the sheriff’s wife in a murder for hire scheme.