We have three parrots in our apartment. No one seems to notice them unless the balcony door is open. Don't think the landlord minds, but I think it helps that we've had the apartment so long that he's the third or fourth landlord to own the place since the original lease was drawn up.

The rules here have come & gone on dogs but afaik anything else has always been ok. The problem with dogs has been their crappy owners. You wouldn't believe how many people have rented in our complex with dogs and then proceeded to let the dog destroy their apartment [scratching & eating up woodwork & carpeting everywhere], or refused to take care of them [a few years ago this one tennent kept their dog in a crate so small it couldn't turn around and then left it like that on their balcony to bark all day while they were at work]. Then there have been all the dog owners that wouldn't clean up after them and things like that. It was a big problem. The irresponsible dog owners greatly outnumbered the renters who were responsible about it.