I know that the Register Mod will prevent animals, and also the OverWatch Mod will stop pets from being spawned for pet adoptions. http://nraas.wikispaces.com/
It almost sounds like those babies animals are being spawned for the Pet Adoptions, so OverWatch will be the one that will put a stop to your problem. I use to have this very same problem as well, until I used this. If you do use this mod, you will need to change the settings for it to take place.
◦Stop Pet Adoption ◾ Whether to disable EA's Neighborhood Pet Adoption system
◾Default: False ~~~~ You will need to change this to True.