Rather the opposite....the privately owned land is a footpath by virtue it being a road related area.
I'd be arguing that it's a road related area because it's a footpath.
I'd be arguing that it's a footpath because it's open to pedestrians.
Open? Yes, because it's unfenced/ ungated, and
there are no signs (put up by anyone) saying anything like "No walking here"
The thing it is expressly not open to to the public for, is parking.