On occasion you can find a trailer park that has a few RV sites, but I suppose you've checked them all. An RV does not meet the building codes for a mobile or manufactured home, so yes the rules for parking them are different.
Sometimes motels will have a few RV sites in the back where you might be able to rent by the month.
Check for seasonal or resort campgrounds in your area. These places will rent a site on a yearly basis and people set up camp there for the whole season. Sometimes there are rules though against living there full time as they don't want the camp to turn into low income housing and want to keep it a weekend retreat for families. If you are both college students and plan to be taking off in less than a year, explain your situation and they might grant you an exception.
See if your local state park or another campground is hiring camper hosts or workampers. You will be required to work a certain number of hours a week, maybe not good for college students, but you'd get a free site and maybe a small amount of pay, which is great for college students.
Another version of the last idea is to just find a convenient campground where you can work out a monthly rate with the owner. State parks usually have rules regulating how long a person can stay, but I've seen exceptions granted in my area.
I just checked www.rvparkreviews.com
and it looks like you have a KOA near the center of Provo, convenient. There also looks like a nice state park west of town, maybe not too convenient but at least scenic. A third choice is located in between those two and sounds as if it has long term sites for rent, though might be a bit seedy.
Best of luck,