Many landlords and tenants might be surprised to learn that under the general law, tenants DO have a right to keep pets.
So if a tenant goes into a property on a handshake and a promise to pay £X rent – there is nothing to stop them from having a dog. Or a cat. Or a rabbit or a parrot. Or any sort of pet.
The reason why most tenants are not allowed to keep a pet is because their tenancy agreement says so.
So the pet prohibition will depend upon the enforceability of the relevant clause in their tenancy agreement.