Read "parents who care about education" as "Children who want to learn" and you've got it right.
As an adult who went to private school all my life with the exception of a state university, I can honestly say that is not the case. I knew PLENTY of kids who didn't give two shits about their education while in private school. The reason they stayed in? Money. Their parents threw money at the school, and the student stayed in. No matter where you go, there will always be motivated students willing to learn, and unmotivated students, who don't care. Private or public.