There is also a pets key. (Looks like it currently has ~1.8k uses.)
This is what the OSM Wiki page on it currently says:
The key
pets=*is sometimes used to indicate whether a feature or venue, such as a restaurant, cafe, mall, hotel, or beach allows pets to accompany their owners or sitters.
Seems like that’s exactly what you want.
So, you’d tag:
pets=yes- Pets are allowed.
pets=no- Pets are not allowed.