mljeckard-
Your willful ignorance and fear of a dog, showing no signs of aggression, at a dog park and therefore most likely not an aggressive animal, is probably not a legal justification for your drawing your firearm in city limits.
Your waving a firearm around in the dog park might, however, be legal justification for your being shot, depending on what, exactly, you're doing at the time.
People with concealed firearms do need to remember a few things:
1. You're not the only person in the world carrying a gun.
2. Someone who draws a gun not clearly in self-defense is a lethal threat to innocent people and is risking being shot, without warning.
3. Doing stupid things with guns is bad for all of us who support RKBA, and therefore doesn't merit any sympathy.
You refer to "pit bull" as a breed. However, no such breed exists. At least learn a little about dogs, since you clearly fear them -- and learn something about some breeds that have different-shaped noses but can pose a greater threat to you.
Reading dog communication is also a good thing: I was once (stupidly) almost bitten by a Golden Retriever because I ignored its body language, whereas I was also charged by a Doberman at full-speed -- he stopped, leaned on my leg, and whimpered until I scratched him behind the ears. My "pit bull" -- a dog born at the pound, not a papered dog of any sort -- will sit in the back of my Jeep, grin, wag her tail, and beg people to pet her -- many do, but some are afraid of her just because she looks like "one of those dogs". My Vizsla, OTOH, a beautiful purebred hunting dog that nobody seems worried about approaching, would likely attack and injure you badly if you stuck a hand in.
If someone is afraid of dogs, any kind of dogs, it's good to learn something about dogs -- not breeds, but dogs. All dogs are the same species, and warrant similar respect and caution, especially if you don't know the individual dog.
Furthermore, a working bull dog of any breed (you will probably NEVER see one in a dog park, and seldom out in public), as well as some dogs like Catahoulas, are bred to ignore pain. That means you'd better know how to shoot to stop. Wounds -- even ultimately fatal wounds -- may not stop an animal that doesn't shrink from pain. And some working shepherd dogs can run across a street, leap, and have you flat on your back bleeding, before you could unholster a gun. If you can't anticipate that, you're SOL.