If you want a parody, try The Day That Santa Died.
"Down In Yon Forest" is in the Oxford Book of Carols, as is Tomorrow Shall Be My Dancing Day; there's a wonderful arrangement of Good Christian Men in the OBC. I like the minor melody carol in the Quaker hymn book, to the effect of "She's black and she's white and Muslim and Hindu and Jew, he's (I forget)... and, oh yes, the Christ Child is you".
Other ones I haven't tired of yet: Christ Child Lullabye, more later.
Blessings,
Barbara