It seems odd to me than an after-school program has no set way for parents to contact the program without going through a relay-style process. I agree that the solution shouldn't be "give out your personal phone number," but it also shouldn't be "rely on a person that's not at the program to relay calls to you."

Put another way, how would a parent contact the program in an emergency? They'd likely (as you illustrated) go through an intermediary that may or may not be there. That seems less than ideal, and certainly wouldn't be something that I'd be happy about if my child were in the program.