A private school is a school which is funded by non-public monies. In other words no government funding or other tax-payer funds are used to sustain the school's operations. If the school is incorprated and established as a 501 (c) (3) entity, it will generally not be liable for local and state taxes. In that sense a private school is subsidized by the public treasury. For this reason many private schools consider it politic and prudent to pay property taxes, sewer taxes and other local taxes to ensure that the local services such as fire, police and emergency first responders are available when needed.
A country day school is a nice name for a private day school set on some beautiful treed acreage. That's right. It is really nothing more than a marketing term.
A parochial school is a type of private school. Generally parochial schools are attached to a church or other religious institution. That institution usually subsidizes the operations of the school as part of its ministry. A parochial school is usally governed by individuals selected by the religious institution. In the Roman Catholic Church parochial schools are frequently operated by the larger institutional entity known as a diocese or archdiocese.