The parents should pay a good amount of taxes, especially a strongly progressive income tax, the government should take that money and provide free education from the nursery to the first university degree for every child, no matter if that child is dirt-poor or born with a silver spoon in it's mouth.
Same system works for health care too, by the way.
What about kids going to private schools?
I don't see the problem?
If parents think that free, state provided public education is not good enough for their offspring, they are free to pay for private schools. As long as they pay their taxes to keep the public education system going, they have done their duty.