Hey hun,
That's really not of fair of your manager, I agree stick to your guns. I don't think the mother is at fault, schools often leave it to the last minute before they tell you things but obviously if she knew earlier she should of asked before now.
If the manager can't arrange cover from another source, she should of said no to the mother and not even put you in that situation. When your working mother you can't expect to have it all, there will be things that you miss maybe the mother could of asked someone else to watch her child during sports day.
Also its not the same booking off a week and going in one of the days, its wouldn't be much of a break for you and I bet your manager wouldn't of changed their own plans xx