Not according to the dictionary: http://dictionary.reference.com/browse/parochial+school?jss=0 (See also http://www.thefreedictionary.com/parochial+school and http://wordnetweb.princeton.edu/perl/webwn?s=parochial%20school
Yes, one of the definitions does say "especially Catholic day schools", but the general definition is a Christian school, as the other dictionaries back up. Just because no one you know would call them parochial schools doesn't change the fact that that is what they are, by definition.
I really wish you would have double checked before posting. Even if you "know" you're right, it's always worth double-checking. It's something I'm always careful to do before I post, because I don't like being wrong. Normally I don't bother to cite things that only take 15 seconds to look up, but apparently it's necessary here.
As far as parochial schools being a subset of private schools, I'm not sure what the confusion is. It's a pretty simple statement; parochial schools are a type of private school. This is also by definition.
I'm continually astonished by posters' ability to turn anything into a controversy. This isn't a matter of opinion; this is matter of dictionary definitions. You may not like the definitions, but they are what they are.