Kids are smart, but they are barbarians. You really think they're going to listen to what their parents tell them to eat when they aren't there to enforce it?
Have you ever been a kid?
Nope i certainly don't think that and most of times i still am.
It wasn't about kids buying the 'wrong' food. It was about parents being an important part of the eating habits a kid creates. Or takes over probably in the problematic cases. Kids buy the food they like, it's up to parents to make sure they also eat enough of the 'right' food in the first place.
And my 2nd post was a direct answer at the bolded. Parents (should) have big influences on kids behaviour. Otherwise... bad parenting. (most of the times)