Tolkien and Parody

[Fordim Hedgethistle]

I don't know if this constitutes an answer to your question Esty but Tolkien wrote what could be considered a parody of LotR himself:

"The Lord of the Rings
Is a funny old thing;
If you like it, you do,
If you don't, you go 'boo'."

Not exactly a fully developed self-parody, but it does make light of his work in a playful manner.