"The Park" is a Restricted Byway, which gives a legal right of way to pedestrians, horse-riders, and non-motorised vehicles. See https://osm.mathmos.net/prow/progress/suffolk/west-suffolk/great-barton/ . There may be other rights to, e.g. if it was also a Public Highway.

However, the street does not appear to be listed on Suffolk County Council's List of Streets (https://www.suffolk.gov.uk/assets/Roads-and-transport/road-adoption/2021-08-03-West-Suffolk-List-of-Streets.pdf), so it's probably not got any further rights. Hence it's likely to be a private road as far as motor vehicles are concerned. In OSM terms, that's probably motor_vehicle=destination.

It's already tagged with motor_vehicle=destination, so routers shouldn't be sending users along it unless it's to get to a location on the road. Is there anything else that needs checking/updating here?