"biri" means "someone", and "herhangi biri" means "anyone" as in one person (3rd person singular). This is true even when we are saying "one of you guys" (biriniz) or "one of us" (birimiz). So the question is literally asking "Is any one of you going to the park?" Maybe several people will say yes, they are going... but grammatically we are asking a singular question "Is anyone going?" Therefore, mu is singular to match "biri".
However, if we were asking "Are all of you (hepiniz) / some of you (bazılarınız) going to the park?" we would use "musunuz".
I hope that makes sense :-)
As the previous responses clarified; the question's structure here yields an expression of a singular entity ("any"one) rather than a plural one ("every"one). Thus, you should use a singular "question word" rather than a plural one... So, we are left with either "mu" or "musun", but then, the entity in question being an individual, and you are asking a group, i.e. he/she doesn't represent the group in question (which is a second person plural), thus he/she must be referred to by a (third person singular) interrogative particle which is "mu" in particular. I hope that makes sense, although I'm not an expert; I'm just trying to improve my understanding by explaining!