For one there is no indication that parents are any more literate in regards to digital practices than their kids. More importantly the constant appeal to the responsibility of parents misses that this is a collective action problem.

The reason most parents give up to regulate their children's online activity is because the children end up isolated if an individual household prevents their kid from socializing online. All the other kids are online, therefore switching individually ends in isolation. What might be beneficial for each household is unworkable as long as there is no collective mechanism. (which is the case for virtually every problem caused by social networks)