I have serious qualms about home schooling standards and would like the ability to home school to be strictly monitored, essentially having permission given only a situation where there is some clear impediment to the child attending public schools (preventing parents from keeping their children at home to brainwash or because they are overly protective). Private and parochial schools are fine with me as long as the latter get no government funding, and public education is the best thing since sliced bread.
Why allow private, but not homeschooling?
Private institutions, because of the number of students, tend to average to some level of decency. With homeschooling, you're more likely to end up with nutcase parents doing the teaching. In any case, private schools are slightly regulated (though to a lesser extent than public schools).