Especially, what my father explained to me (he directed projects and extracurricular 'homework help' (+ free food) for kids in the poor neighborhood of my city at the time) was that the most important for success in school was parental implication, and that if you compared kids in 'regular' school with present and interested parents (often children of teachers, or children with one stay at home parent and few siblings) to kids in 'special school' (Montessori at the time i think), there was no practical difference on their success later. He told us his job was to try his best to have his street educators bridge the gap between parents and school, and if it wasn't possible, to offer a more collective and affirmative way of learning/doing in his extracurricular center.

(weird translation, sorry if it isn't understandable, there is a lot of specific vocabulary i tried to simplify then translate).

It was only his impression, but I've worked with children from my 14th birthday to my 25th, and my personal experience (i taught science through experiment with an association both at public schools and at summer camps) tends to confirm that what's really matter in engagement from the parents.

This has not been my experience at all.