I think it depends on what you're used to and where you live. In Alaska I've read the kids go out and play in below zero with no problems. I was baby sitting my cousin's kids in St. Louis, and on the way out the door dad said "It's only supposed to be 11 degrees today, so don't let the kids outdoors until later in the morning." And one of the kids actually fell through the ice on the koi pond (it was off limits) and got wet, but they snuck in the house and didn't tell me until they were dry! LOL
Cold air, per se, doesn't cause problems, unless a person has a medical condition. Don't know of any lung conditions that are made worse by cold air, but if a parent doesn't want a kid outside for a particular reason, and has put it in writing, the schools should respect that.