Assuming that your families are merely paying to have their children be eligible to play, and that they are not “members” who have the right to vote for directors or on other significant corporate action, I see no reason why you cannot limit your customers/participants when you know that one member of the family is obnoxious and disruptive.  Even if you are so large and non-selective in your admission procedures that you could be considered a “public accommodation” and not a private club, you would be restricted only if your discrimination was based on something like race, ethnicity, or some other protected class.  Obnoxious people are not a protected class.

If your families are members of the corporation who have the right to vote, you may be required to provide them the opportunity for some sort of due process hearing to expel them from continuing membership.  (See Nonprofit Issues®, Vol. XXV, No. 5, 2015)  That procedure would normally be spelled out in your bylaws, and good bylaws will permit you to deny admission for any reason whatsoever.  (See Ready Reference Page:  “Bylaws Function as ‘Constitution’ of Nonprofit Corporations”)

You may not want to punish the kid because of an unruly parent, and you may want to develop a type of probationary participation that can be terminated immediately upon any further harassment.  But if you think enough is more than enough, I don’t see a legal reason you have to let the kid play.