I am an easily irritated person. To that end, whenever I get irritated, I ask myself one question. "Why does it matter?"
If you are honest with yourself, the answer can tell you a lot. Plus, it helps you to find solutions instead of just being cross.
So, why did it matter? If it's purely that he didn't do as you asked, I say get over it. You aren't his parent, and it won't hurt him to not go to the park. That's a battle for his parents to fight.
If it's that he's a loud, disruptive child, then I'd offer him a choice of a few quiet activities, but not insist that he goes to the park.
If it's that you were relying on him to watch the littler ones, then you should be doing that yourself.
It sounds like you really don't like this kid, which is fine, but then don't babysit for him. It's not good for either of you.