For the sake of this metaphor, all of that would be equivalent to the park, custodians, and park rangers. The latter two are paid by the government (the "owners"), but few people would argue a park as a "commons". So, maybe?
main difference ofc is that few pay for a private server, despite contributing to it. Parks are paid for by taxes, and sometimes voted upon by citizens to allocate budgets for. But a park with no visitors is similar to a forum with no visitors.