This seems like a question to ask the Town. Just looking on Google Maps, it looks like an old right of way, where Quaker Rd would’ve run through all the way to Rt. 120. This might still be a paper street, where the Town stills own the property, but the road just no longer exists. Residents may or may not be aware of it (which could lead to disputes). Alternatively, residents may have petitioned the Town to abandon it, and then it would all be private property. But if it’s cut and maintained along the route, then that makes me more inclined to believe it’s still the Town’s property, that’s who I would ask.