How to determine if a noun is a place or a thing

Ah, this stems from teachers telling students that nouns represent things, verbs represent actions and so on and so forth.

Unfortunately, none of this is true. It's a seemingly handy generalisation for helping students intuitively identify nouns before you actually get down to discussing nouns with them properly. Otherwise it's pretty misleading.

Nouns often represent actions, for example, baptism, or massacre. Verbs often represent states and not actions - for example the verb have.

Nouns are words that typically represent things, but may not. They are nouns not because of what they represent but because of how they work in the grammar. They tend to inflect for number, for example using the suffix s. They often occur with articles and determiners. They (or the phrases that they head) are very frequently used as Subjects, Objects, Complements of prepositions, Modifiers of other nouns. It is these things that make nouns nouns, not what they represent.

Individual words may represent both places and ideas at the same time, or perhaps things and ideas. What exactly a noun represents isn't really tightly fixed in the way we imagine it to be. It can change radically or subtly with the context of the sentence that we see it in and the context of that sentence within the larger conversation.

The upshot of this is that if a park seems like a place to you, it is! If it seems like a thing to you, it probably is too. It depends on how you're thinking about it. There's no 'right answer' to this question. You are free to decide as you think fit. Freedom is a wonderful thing.