I agree with @LuckyGuy. I say the very same thing about people eating in their car. How far are you driving that you have to constantly be eating and drinking. In fact, I think the same about movies. During those two hours you have to buy a popcorn and coke that is 6 times what it should cost? I understand for some the food at the movies is part of the whole experience, but I am just too cheap I guess.
But, back to the main question at hand. I think parents should pick up or have their child pick up after themselves. Food left behind can attract a pile of ants or other insects, let alone a child might be allergic to something. If someone left a piece of food behind in a grassy park I probably would not be so strict, but in a playground anything left on the ground I would consider to be litter. Imagine if lots of kids left food behind.
I guess if your child is extremely allergic, in the end it is the parent of that child who must be extra vigilant, because you can’t rely on any area to be safe, you have to check for yourself.
But, seriously, the food is out of control. We almost never ate while playing outside on a playground when I was a child. I can’t remember doing it at all, but I guess I might have now and then and don’t remember. A piece of candy maybe? We went inside to eat. Or, we sat down on a bench and ate a snack, and threw away all of our trash.
FYI: Someone mentioned above planes not having peanuts on board, and there still are some airlines that do. Delta for one almost always has peanuts to offer in afternoon and evening flights.