Dogs successfully living with other pets (cats, birds, ferrets) is a topic that quite often comes up in my conversations with clients.
At its core, teaching a dog to not chase small animals is no different than teaching an off-leash recall.
It is impulse control. The more you practice it - working on it in all kinds of environments - the better your dog will become at dropping the chase, looking up to you for guidance, and responding to “leave it” immediately.
This teaching and proofing phase can be tricky, but I find that the harder part comes later. It is the part when people think that they “taught it.”
No matter how good your dog’s impulse control is or how laid back they generally are, at their core, they are still superb predators. They have a set of large teeth, growing from powerful jaws that can crush bones and tear ligaments. Their bodies are equipped with strong muscles that allow them to move swiftly, to jump far, and to maneuver with grace. Most importantly, their prey drive does not go anywhere just because you taught them how to suppress it.
This means that when circumstances are “right” - your dog was overwhelmed, or irritated, or experienced something that affected its ability to effectively control its impulses - they will snap at that flying parrot or running cat. If you were there to control it, chances are that nothing bad will happen. If you weren’t - no words are needed.
Cats are generally smart enough to not bother unsupervised dogs. However, some other animals, such as small but stupidly bold parrots, do not have this foresight.
Regardless of what kind of small animal cohabitating with your dog you have, don’t let unsupervised interactions to happen. It is honestly not that hard. Tragedy can happen within seconds; seconds which you cannot bring back.
So can dogs coexist with small animals? Yes - if their owners are smart about these interactions. No - if they are lazy or believe in letting animals figure it out on their own.