The real question is does the parking lot of a school administration facility qualify as a school in the eye's of the SDA? No students attend classes in this building.
I find the notion that it is primarily a church which has a school as a secondary function to be pretty persuasive. Also, other private schools in the area have school zones defined on the streets that border them; this one doesn't.
That said, I think that any law that bars otherwise legal activity at "any public or private school" would eventually run into Free Exercise problems if it were pursued far enough. Same for the law barring permit holders from carrying outside the vehicle on any college campus, public or private. I don't think it's outside the realm of possibility that there are "real" private schools where the local board would at worst not mind and at best encourage responsible parents to carry concealed at school functions. i.e. why can/does the State regulate weapons carry at a school run by a church, but not at the church itself. Surely there's Sunday school going on there...