You're barking up entirely the wrong tree.
I think you need to take my facetious comments less seriously, step back and and see them for what they are.
Don't take them seriously ... they have nothing whatsoever to do with parody per se or the quality of the particular piece. I know very well what the original was that Stephen’s piece is a parody/paraphrase of, I'm just trolling him about an entirely different matter: namely, that it is not now Christmas ... and that his decision to post this now means, therefore, that his timing is poor (he should have posted it at Christmas instead).
I'm simply (entirely arbitrarily, just because it is possible to do so and for no other reason) suggesting that the joke might be fine (it is ... very), but he's still a lousy comedian, because he doesn't know when to tell it (it’s the way he tells it that’s the problem).
That's it ... all there is to it — it has nothing to do with parody, satire, anything like that ... and everything to do with Stephen himself.
All it amounts to is “Boo! Get off!” heckling, like you’d do to a friend on stage.
I can't make it any clearer, I'm afraid.