It would sound extremely strange, "at" being used to describe static position. We can say "She's at the park" to say where she is, but running involves movement around a space or area, in which case we use "in". I would never say "I've been walking at the park", for example, always "in".
"Extremely strange" might be an exaggeration. In my opinion, "We rarely run at the park" could mean running close by or in the general area by the park. If you switch out "Park" with "School", the difference between "Running in the school" and "Running at the school" becomes quite clear to me. The first one to me means running inside the actual school building and the second one being on or around the school grounds. But that's just my opinion :^)
You are perhaps thinking of Latin/German, where motion towards something takes accusative and motion from takes ablative/dative.
In this case, the motion is happening in a place, indicated by "w", and is therefore locative (a case relating to a location or place) is used
Actually, i was thinking about another slavic language, the russian language, like in this example: https://forum.duolingo.com/comment/11856389 In which the accusative describes motion of the verb побежать, associated with the preposition в, that i was assuming it would work like "w" in polish in front of a verb of motion. You can see it takes accusative in russian. But that might be not the case in polish, i suppose.
Indeed "w" can take accusative, just as "в" can take the prepositional case :-) . The "в" in the Russian example has the children running to the park. In the Polish example, "we" are already in the park when we do our rare bit of running, hence "w" + locative.
"Rarely do we run in the park" This is the correct translation according to duolingo, at least in my exercise. Could anyone explain me why this "do"?
I agree with this, that 'the correct Polish is more important'. Actually, quite often when I try to think of something in Polish I remember it better by first thinking of the more literal form in English. For example, with the dative case it helps me to add (in my head) words like 'give' before the verb 'help'. Thanks for the reply!