I don't think it is parental 'cleverness' that has the most impact, but rather parental 'interestedness'!
Certainly here in Birmingham, competition is pretty fierce and from what I have seen, laid back parents have, with very few exceptions, little chance of their child going to a KE Grammar.
Parents have to be fully on board, interested, supportive and, admittedly, a little pushy, to gain success.
I have never heard of a child who has casually wandered into the exam and then passed for a KE Grammar. This is because of the competition and amount of prep so many children are doing.
As for Grammar school parents=Grammar school children, this seems to be the case. At parents' evenings, quite a few of the parents are old boys (and/or old girls of the sister school). My son says quite a few of his friends talk about how their parents went to the same school. Indeed, one was taught by the same venerable teacher his son has, who is now 74 years old!
Even now at 12, my son has said a few times that he wants his children to go to the same school as him