I came across @Randal'Thor's creative "edit summary"
ring a ding dillo, hey Puzz, merry Puzz, Work a place illo
on this question and it got a chuckle out of me, but also got me thinking whether there is any technical term for this kind of writing.
- The structure is clearly intended to point strongly to the original work, in the case a song sung by the character Tom Bombadil from Tolkien's Lord of the Rings.
- It is derived only by modifying certain phrases, and not wholesale rephrasing or rewriting.
- It is not (necessarily?) intended to mock the original work, but it instead takes the original work and applies it to a new situation, or extends the scope of the original work. I'm not sure if this is a characteristic feature, though.
For some reason, my brain is telling me that there's a word/phrase that describes this kind of writing, but I can't find it. If I can find more examples describing what I have in mind, I'll add them to this question.
Rand suggests that the word I'm looking for could be "parody", but I'm not sure. Perhaps there is something more specific.