I'm wondering if you're not originally from the UK HadaVerde?
I have only started using the word playground to mean a public space for kids with swing, slide etc since I moved to Germany because I now have friends from various countries who understand something different by "park". When I lived in the UK I would always have called that a park, playground to me meant the paved part of the grounds of a school, whether or not it has play equipment in it (most schools didn't where/when I grew up).
I don't have a general word for a large public open green space - I would refer to a specific one by its name, for example I gew up near Leamington Spa and we had Jephson Gardens, which is what we called it (Or just Jephson's). Within Jephson Gardens, there are two play areas, which I call parks. So I might say to my friend "I'll meet you at the big park in Jephson Gardens".
I think this is fairly standard usage in the UK, I've never heard anybody using these words differently. It was only when I moved out of the UK that I realised park means an open green space somewhere else and the word for a place for children is playground or play area.
Parks (play areas) are generally public and anybody can use them unless they are specifically gated inside a private area. IME most people have a variety of local parks they visit, and either the one closest to their house, or a biggish central one near their house is the one that they tend to socialise and meet friends at. That doesn't make it exclusive for their use as it usually is on several other families' rotations.
Kids like variety, they'd get bored going to the same old park every time.