Regardless of what your forms say in terms of how long a parent has to pick up they get there when they get there. There are very few jobs a person can just walk out the door on and leave. Not everyone is the top executive and making all the rules. Just as if the school called to say your child needed picked up and it was the middle of naptime and you had no vehicle and child normally rode school bus. You would not be able to get all the kids up and walk to the school in the hour very likely either. I get that it is frustrating but that is also what a day in the life of a typical large family looks like. It was also a perfect opportunity to teach the other children some empathy in the sense of I know you want to go outside right now but child XXX is feeling too sick. Let's leave her to rest and I need you all to play over here quietly or whatever.
Also the parent should have given you a heads up on their situation and advised you to move on to the contacts - although in most cases they are also working family members and certainly would not be able to leave their job to come.