It depends on the provisions of the document establishing the easement. That document may also establish the County's responsibilites. The other possibility is that your access is through a road that was platted between the county park and school property. A look at the plat map would show this. Once a knowledgeable lay person or a real estate attorney reviews the relevant documents, he/she can advise you about responsibility.
The advice given is general for information purposes only and should not be relied on. You should consult with an attorney of your choice to fully advise you about your legal rights and obligations.
I agree with the previous answer, and would add in San Juan County costs based on prorated use is an established precedent. It may be that the change in use will be a surcharge on the easement and should be addressed sooner rather than later.
Only If and until you and I sign an Agreement for Legal Services, I am not your attorney. These answers are provided for informational and/or novelty purposes