That gate is https://www.openstreetmap.org/node/4384031961, by the way.

Apologies to jump into the discussion, but my main reasoning there was: if there are multiple gates, opening hours might differ per gate. This can be the case for very large parks (I've seen it in the UK), for example. Adding the opening hours to the park could work for simple cases in my opinion, and it will definitely be the easiest to understand for end users. However, it might not prevent routers from routing through the park when it's closed (?)

I wouldn't personally tag opening hours on paths. The hours aren't actually a property of the paths, but rather of the gates (when they are locked). I think it's also less easy to understand for end users.