I've only once had the problem that someone from the school used their personal mobile to call me and didn't get through, but I was already calling to let them know I'd be late.
There's no reason for parents to be able to contact you socially unless you invite it, but surely you should contact them from a school/shared number?
I'm suggesting this not for the sake of others, but for your sake. Given that people will block unknown numbers, I would think using a known number makes your life easier.
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.
Now, there are other ways this could work. My company has a "director of client services"—let's call her "Anna"—and all parents have Anna's number. So I suppose I could have called Anna, and Anna could have called the parents, and then Anna could have called me back to relay what the parents said. It just would have taken longer.
Of course, Anna is occasionally sick / on vacation / otherwise unavailable, in which case there's a second person—let's call her "Vivian"—who I can reach out to in an emergency. We're an after-school program, so we're not set up to have a centralized office phone, but I guess parents could add Vivian to their contacts as well.
Basically, I'm happy I was able to just call the parents.