Many states limit deadly force to humans-only in such a case (in a public park, not out on the farm shooting coyotes). There are, however caveats:

(1) Proximity -- If a dog or human attacks your dog, attached to you via a 6' leash, a reasonable person would conclude that the aggressor is not suddenly going to become gentle and charitable once it finishes killing your pet: you're next on the menu. Of course reasonable exceptions exist: you might have a tough time explaining why you shot a Chihuahua.

(2) Jury Nullification -- The law is written at the state level, but a rural jury might (and has, on occasion) decided otherwise. For instance, a state may allow lethal force to prevent arson against an occupied dwelling, but a rural jury might decide that a barn full of horses qualifies. Personally, though, I feel deadly force to prevent arson should be considered justified regardless of the structure involved, since it places responding firefighters, police, and EMTs in danger.

Most places have a lower standard for defense against animals, but I think the shooter in question should be held to the higher standard (deadly force), seeing as how a reasonable person could conclude that firing a pistol in a public park could injure a bystander.

In either case, though, the conclusion is the same.