The most important responsibility of parents is to make sure their children grow up in a wholesome, moral society. This is because the influence of parents is highly limited-even in a society of ancient times where the role of parents was highly respected. Nowadays, the role of parents is thought be negative.
הרבה שכנים עושים The priest says to the wayward woman "Confess your sin for neighbors (the general society) have a huge effect. "
\In a similar vein, one might be in a wholesome moral society like a Litvak yeshiva. The responsibility would be to stay within that context even if the parents think that for themselves a different context would be preferable. In fact, this fact alone shows how right the Gra was.
Parents might be highly motivated to do the best for their children, but if they do not know this principle, they are likely to believe what is good for them, is good for their kids. They do not stop and think what kind of group they are in already that is decent and wholesome.