I'm going to refuse to leave the dog park and ruin a meet-up
Why aren't you going to give space to the meet-up?
I'm a dog lover and I don't want to leave the park with my dog.
While you have the right to be in a public space most of the time, you don't have the right when it was specifically reserved for an event to which you were not invited. You could be cited for disrupting the peace by not following orders by event organizers, so you need to suck up your pride and let them have the park for a day.
I don't care.
I really hope that you will be willing to let the event proceed.  As much as you want to use the dog park today, it is not worth getting a fine over.  Your dog will likely also be upset by all of the angry people around.