There is, of course the Great War piece, parodying a Christmas recitation (Christmas Day in the Workhouse)rather than a carol:-

CHRISTMAS DAY IN THE COOKHOUSE

It was Christmas day in the Cookhouse,The happiest day of the year,
Mens' hearts were full of gladness, And their bellies full of beer,
Then up stands Private Shorthouse, His face as bold as brass
Shouting "We don't want your Christmas pudding you can stick it up your..


It was Christmas day in the harem, the eunuchs were standing 'round
And hundreds of beautiful women were stretched out on the ground
Then in walked the big, bad Sultan and gazed on his marbled halls
Saying "What do you want for Christmas Boys?"
And the eunuchs answered...
sung)Tidings of comfort and joy,
Comfort and joy,
Oh, tidings of comfort and joy

Good luck

Walrus