Admittedly my understanding of Virginia's laws are somewhat limited as I have not been CC'ing long. But...
In VA:
1. If you did play a role in escalation, then it is difficult to subsequently claim self-defense.
2. If you did play a role in escalation, but you subsequently retreat and disengage, then it can become easier to claim self-defense. Basically any role in escalation means you have an increased responsibility to retreat.
As I am in very poor shape and have 3 minor children to worry about, my opportunities for a fast retreat can be somewhat limited. Therefore, I make an even bigger effort to avoid, apologize, deescalate, and retreat slowly.
But in whatever state you are in, and no matter what the "Stand Your Ground / Castle" laws are in your state, it is my belief that any time you show your gun, draw your gun, or fire your gun, you place yourself in legal jeopardy. If this situation had continued to deteriorate and shots became necessary, the BG's girlfriend would doubtless claim you were the aggressor he was totally innocent. Other people in the park may not have even been paying attention and probably are not going to be helpful. Maybe you will prevail legally, maybe not. Heck he might even have a gun also. Do you really wan't a full gun battle? What if he is that one in a million "bad cop"? Who is going to be believed? You or this dedicated servant of the people? Wouldn't that make great evening news "Decorated Police Officer Viciously Gunned Down By Crazy Concealed Permit Holder in Dog Park"
Generally a bully (human or animal) has the objective to intimidate and dominate not actually engage in a physical altercation. Your objective is (or should be) to get you and yours home safely. Against a bully, submissiveness is very strategic. Think of it as the ultimate "concealment". Tactics win the battle. Strategy wins the war.
That having been said, I do like the idea someone else had of the wife basically screaming, "Please God, Don't Kill Another One." Not flashing or threatening exactly. LMFAO.
Note:
Against a criminal or predatory animal, submissiveness may have exactly the opposite result. A predator's objective isn't domination, it's dinner, you being dinner. Your objective remains the same, getting home safe. A more assertive stance may have deterrent value. Telling the difference between a bully and a predator could be problematic. It is my belief that most predators are ambush predators, while most bullies are social/situational.
Just my humble opinion. Please feel free to disregard.
Oh, by the way, responding violently to an overly excited puppy may be barely tolerable in your own yard or on the street, but in a dog park it really is over the line. You are asking for some real legal problems if nothing else.
Sorry for the long post.
It may be better to be judged by 12 than carried by 6, but hell I'd rather just walk away.