Greer is not listed as a Metro Park on their website.
Nashville.gov - Parks and Recreation - Park Locations
Metro Nashville Parks are all off-limits, are not posted, and I don't think they have to be posted.
Municode - Search
13.24.440 is the text on their ban in the parks.
TCA 39.17.1311 (d) says they can make all parks off-limits and it makes no requirement to post.
TCA 39.17.1311 (e)(2) says they have to post, but I seem to recall that certain municipal codes were "grandfathered" in and don't have to. Nashville and Knoxville come to mind. Does anyone know more about that?
TCA 39.17.1311 ©(3) looks like the city can simply vote themselves out of the requirement to post.
I didn't stay at a Holiday inn Express last night, so this analysis is probably worth what you paid for it.