My pets only get sick when I change household. I have two I'm playing, all the same family, had to split and take a dog when the eldest's time to marry came up and it was totally full.
It's incredibly aggravating that they've allowed for rotational play and yet EVERYTHING resets, relationships, pet health, inventories, work uniform...it's ridiculous, no way you can actually progress (not that there's much to progress towards!)
But mega sick of all of my pets being poorly at once and being able to do nothing about it because the game refuses to leave that household as it was and make up its own stories.
Don't get me started on the absent mindedness of ALL of them -.-