I think it will depend on who owns the building or what leasing arrangements there are (if any).
A typical lease for a PC owned building will be for the PC, as landlord, to be responsible for the outside of the building, whilst the internals are the responsibility of the lease holder. In the case of a leased out building, the management of running the building will probably be down to the lease holder.
I am sure there are many other possible arrangements, such as joint responsibility for management with the local church or other local organisation.
I would write a polite note to the PC Clerk and ask for the details.