Even if it's private, how is that different from any other private business with a no-gun sign?
It's not a problem unless you're asked to leave, you don't, and the police arrive.
Am I missing something?
No, it isn't any different.
The point is, if it is a PUBLIC park, they shouldn't even be allowed to go that far, putting up signs, that is. If it is a public park, all permitted carry should be allowed, including open carry.