To reiterate, without knowing about your application, it isn't easy to make recommendations.
It sounds like the Animal table is going to be the join table between Pet and the different types of animal tables. Correct?
You haven't mentioned a need to store differentiating information - e.g. species, genus, family - in the different animal tables (Dog, Cat, etc.). So it sounds like you could skip the one table per animal notion and just have the Animal table contain animal types like rabbit, dog, etc. Then it becomes the "one" in the one-to-many relationship with Pet.
It sounds like the Animal table is going to be the join table between Pet and the different types of animal tables. Correct?
You haven't mentioned a need to store differentiating information - e.g. species, genus, family - in the different animal tables (Dog, Cat, etc.). So it sounds like you could skip the one table per animal notion and just have the Animal table contain animal types like rabbit, dog, etc. Then it becomes the "one" in the one-to-many relationship with Pet.