I'm in a pet friendly coop, and our House Rules mostly cover dogs, but I know we have owners with cats and parrots. The rules are basically what you would expect - your dog can't be left to bark all day long, keep the dog on a leash at all times, don't let the dog in the landscaping (our board is particularly nasty and fined a shareholder who distractedly didn't notice that her tiny little dog peed on a fence rather than just sending a notice to remind the shareholder of the rule), owner is financially responsible for any damage your pet causes the coop, etc. We are expected to register pets iand have an emergency contact on file and dogs have to be up on their vaccinations.
I heard there was talk maybe a decade ago about going pet-free, but people who buy in to a pet-friendly coop would be more likely to want the option of getting a pet in the future even if they don't currently have one, and I think making that change (which I guess is just a house rule) would probably lead to board members who voted in favor of changing the rules being voted out.