Or, in more practical terms, why is "no anti-war protesters in the [private] park" any different from "no black people in the [private] park?"

In even more practical terms, where do you find private parks, and who would want to hold a protest there? The exclusion of blacks would violate laws against racism, I guess.

If your trade union holds an assembly in a location they own or rent, they can kick out people who want to voice dissent. If they hold an assembly in a public place, they must tolerate dissenting political speech, but not attempts to disturb the assembly by violence. Why is the difference so difficult for you to get?