Here my two cents; after I made a bit of searching, and found this piece on the site Hinative:A square is more of a social gathering place, Like time square in new York city, where as a park is more of an open grassy area, often with a playgroud for kids to play on.https://hinative.com/en-US/questions/463433
I would add the fact that park, more often than not, is a wooded area (área arborizada/florestal), and in general larger than the former.
But sometimes a square could be a big open space too, so some area of intersection in meaning. But the latter is more prone to have big walks, paved runways, plenty of wildlife, there is - an ampler space in general.
Squares are in general near the city movement, while in parks you can escape from the city noises - traffic congestion, the works!
Most squares in the UK had origins in gardens, open spaces/open gardens where one could have social gatherings. Thus, the first square in London was
So, one can find: Leicester Square, Paternoster Square, Soho Square, Trafalgar Square, Parliament Square, Granary Square etc. You can find them on images from search engines like Google/Bing/Yahoo.
Now, you search for images (or just search) with "Ibirapuera Park" and "Central Park" and you see that are big Urban Parks.
Then, for a change, you can do a Wikipedia search and see the Van Cortland Park:
Van Cortlandt Park's sports facilities include golf courses and several miles of paths for running, as well as facilities for baseball, basketball, cricket, cross-country running, football, horseback riding, lacrosse, rugby, soccer, softball, swimming, tennis and track and field. The park also contains five major hiking trails and other walking trails. Its natural features include Tibbetts Brook; Van Cortlandt Lake, the largest freshwater lake in the Bronx; old-growth forests; and outcrops of Fordham gneiss and Inwood marble.
So, sometimes if you call a Park a Square, this Park will get offended!