In Collins (under US usage) this is the first definition for parochial
narrow in outlook or scope; provincial
the second being
of or relating to a parish or parishes
Merriam-Webster, on the other hand offers this:
- of or relating to a church parish
- of or relating to a parish as a unit of local government
- confined or restricted as if within the borders of a parish: limited in range or scope (as to a narrow area or region): provincial, narrow
Cambridge lists the two meanings as separate entries
- parochial adj OF A CHURCH
- parochial adj LIMITED
As to why, I like Spehro Pefhany's explanation.
On a personal note, during a sermon in which the priest admonished the public to become involved in some action, my 10 year old brother whispered to my father "We're not public, we're Catholic (with a capital C)!" believing the school attendance distinction (we were Catholic school kids, not public schoolers) carried over into all aspects of life.