Even taking a non-cynical look at certain parents' level of interest/ability/care/etc, we go as far as taking students and teachers out of the rest of the workforce for the express purpose of being able to assign them accountability for education instead. There just aren't levers like that available on the parental side to try to trigger meaningful actions with, nor is there anything close to consensus on how those might be put into place to begin the debate on what parents should be made accountable for with them.
The best chance I've seen to increase the amount of involvement from parents in their child's education is to try to have enhanced their own childhood education well enough to see why it's so important they be actively involved.