Hi Joe,
I think the unsentimental tone works well here.
I'd be inclined to lose the ellipses and revert to the original opening. It actually works absent the title, too, anyway: "to the vet with our decrepit cat:". And the lower case does enough show the run on from the title. It's how it's conventionally done, I think. In fact, you don't even need the run on, unless you want it. You could start "To the vet with our decrepit cat:"
Most lines have 5 beats, S1L2 has six, and s1L4 has (arguably) four. I don't know if you're intending this variation or not.
Is the cat itself "epiparasitic"? Maybe I misunderstand the word, but I think that would mean the cat is a parasite of something that is itself a parasite. I could see that the cat might be cast as a parasite on its owners, but who are its owners parasitising? Or do you mean that the cat's parasites have parasites. In which case, maybe something like, "his parasites have parasites" would be better -- the repetition adding more comic effect, and possibility necessitating fewer visits to a dictionary
Though it'd require some reworking.
In S2, "But" is capitalised. Should it be? With the capital letter it reads like the cat's name is "Tipperary But"? People call their cats all sorts of things, mind.
Maybe you could insert another "cat" toward the end of S2 to rhyme with "that" (and "flat"). Maybe, "He's had enough, this cat."?
best,
Matt