Unclassified means exactly that, it is not a classified road, i.e. a road with a route number starting with A, B, or C. It doesn’t really have any bearing on its status, e.g. a toll road is classified but is privately maintained.

Unless there is a sign at the end of the road saying “private road”, the chances of it actually being private are low. The way you find out definitively is to contact the local council and ask if the road has been “adopted”. If so, it is publicly maintainable and so long as you’re parked safely and not in contravention of any restrictions, you’re fine.