It's fair to note that it's not all about intelligence, but then you shouldn't make the intelligence comparison between parents and teachers.

In almost all cases, parents care more about their kids' development than a government employee does. Why wouldn't you expect that to overcome the intelligence gap? It's not like parents have to come up with the education materials themselves (just like teachers don't). They can use just as good of teaching materials, and often better, than the schools use.