Prefix: I know you were rebutting the idea of rate parking at universities. This reply is just to cover some other possible solutions that are not incompatible with the rate parking idea.
As a student who attends a university that traditionally is considered rural (VT has grown significantly in the past decade, much more than the surrounding towns so I don't know how "rural" we are any more), we struggle with the parking issue as well. This is exacerbated by the housing issue however that's besides the point.
Teachers obviously should get priority with parking and with rate exemption unless there is some exceptional public transportation alternative. If teachers could get by without driving to the school that would be awesome however it obviously isn't feasible since you are permanent residents and likely live further away from campus than students since you can more likely afford the beautiful rural properties.
I could see something like teacher parking sections working well if they were allocated based on need in an area and parking spots converted to also allow student parking when the need for teacher parking was low. Depending on how it's done it could be somewhat complex and require some getting used to however time based zoned parking has worked reasonably well for city residents and I couldn't see it being too difficult to adapt to rural environments.
Our campus has gotten parking decently under control recently excluding the issues due to overcrowding. The things that helped were the free bus system allowing students to use that instead of parking on campus or parking away and taking the bus to campus, those infuriating ride scooters also allowed students to park further away making space for teachers, and the teacher parking spots which convert to generic parking after hours.
We have a lot to improve here but things are at least workable considering our situation. Also this is definitely more apt for a larger college town and I recognise it may not apply to a smaller one however there has to be an analogue that can work for small rural campuses.