Or, you could get to know the people who own it. I’ve been invited all sorts of places that are closed to the public. In this case, they’re trying to keep it open to the public as much as they can, within reason.

I see it kind of like if I let someone house sit for me, and then they turned around and set it up as an AirBnB, and some of the guests came in and trashed the place. Even if the house sitter made improvements to the place, it’s not their house. When the owners come back, they’re within their rights to kick everyone out and ask for reparations from the house sitter. But they may decide to work out a compromise instead so that everyone gets something, even if everyone gets less than they feel they deserve.

It’s not a public park then.

Well, technically, all public parks are owned by the nation state(s) which govern them.

You're not allowed to visit a park in another nation without permissions pretty much anywhere else on earth, you should feel privileged that you can visit this one except for specific days or seasons.