I'm Lutheran and have been involved in a number of parochial schools.
Most of them are under the "control" of the local congregation, primarily. They do have ties to a circuit and district, but that's primarily for support and continuing ed. The curriculum, budget, hours, etc. are all set locally. If a Lutheran school is run by just one church, they have a board of education. That board oftentimes also runs the Sunday school, Bible studies, and other congregational education programs. The school is just an extension of the church.
Currently, I'm at an association school. It is run by several churches of the same denomination and synod. They each have a say in the budget and other typical board of ed things. They each send two people to the board of ed, so they are all equally represented. The pastors may or may not be on the board of ed (usually they are in advisory role, but I have seen some take more powerful positions.)
The public school district works with us to provide some services, but we don't even follow their schedule exactly. We share some bus services, but since our students have to pay for that, it doesn't affect too many of the students. The whole busing issue is different in every district. I've seen all sorts of things worked out (or not worked out.)
I've got one student in a Lutheran high school and one at elementary. I've worked at elementaries (K-8) but have helped out at the high school level. My husband (a pastor) serves on one board of ed and the technology committee at the other school.