Parochials can take who they want...deny entry if desired, public schools are forced to take everyone from willing students to felons.
Parochials do not suffer from "PC over common sense" disease
Parochials have an effective discipline program
Parochials can expel students, public schools will undoubtedly face lawsuits if they do so, and will likely end up being forced to take the kid back...rendering discipline meaningless
Parochials are not free..."that which is valued too cheaply is esteemed too lightly" (TJ) - those who go to parochials have parents who value education more, generally speaking - hence parochials have better product to work with from the get-go.
Parochials can, if they're bad, close/go away/out of business.
The problems in public schools are a result of a monopoly system. The symptoms we all see are the results of that monopoly.
Want to improve education? Fix the problem, don't treat the symptoms. Get government out of education; make education a competitive industry.
While I agree with you about the advantages of private schools, I disagree that government is the problem. The problem is LIBERAL government who, for at least the last 20 years, has watered down content to nonexistant levels.
Consider the education system in India. Free to all, and expectations through the roof. Those who don't excel in school become blue collars. The Universities are government schools, and they have higher standards than any in the US.
The answer to US public education? Raise the stakes. If you don't have, say, a 3.0 in high school, you don't go to college. Period. You go to trade school, or work for your father-in-law at the auto parts store. All work in honorable, and everyone can't be white collar.
A college education used to mean something, and it was not for everyone. Now it just means that you (or someone on your behalf) wrote a series of checks, and you showed up for the finals.
Can you imagine the uproar from the liberals if a conservative proposed a 3.0 minimum for entrance to state universities? But that is what it will take.