I am trying to figure out the best profit maximizing way to price an app that will be used by schools. So we could charge the schools or the parents and it could be a one time thing or subscription based or any combination of those things. Are there any other ways to price this?
A/B test accordingly, good luck.
There are a few overlapping issues here. Public schools have an acquisition protocol that requires you to become an approved vendor to the city before you can offer your software/app for sale. That means every single district you'll have to go through this process, one at a time. And then they have to have permission to choose you for some specific requisition. They can't just decide to use something they like because it's available.
If it's parent driven, the school still has to be willing to move resources to interact with whatever portion of the software makes the app useful to parents/students. This might even be worse than the above because now you're trying to convince each individual school instead of all the schools in a district.
Adoption is the tough nut to crack. My feeling is that you'll end up with some kind of freemium model. There are basic functions that everyone will want to use, and this will introduce them to the nice-to-have functions they'll want to pay for. I don't know what your app is supposed to do, but if the free version were to include the lunch menu, all event information, and push notifications from the administration to parents, all as free, then anything else was premium, you'd have a good hook.
I don't know what you've already done to solve the adoption barrier of government regulation. That needs to be addressed before you worry about how much money to charge for specific benefits.
Not sure if these would work with your app, but, if the school pays, you could charge the based on the number of teachers, students, and/or parents. If the parent pays, you could charge them per student.