I suspect the two car parks mapped here as osm.org/node/27538311 and osm.org/way/202689837 are the same and can be combined

I'm not certain, but I think the road from Edge Road to the car park at osm.org/way/228724845 does not exist. I suspect this car park is reached through the car park at osm.org/way/228724840 from the entrance on Bank Road.

The tagging for the bridge compared to the highways either side seems odd. Presumably the same designation for horse and foot applies and bicycles are permitted. This could do with a survey to check.

The bridleway at osm.org/way/378204098 isn't tagged with "designation=public_bridleway" but to the east osm.org/way/378204095 is and to the west osm.org/way/54276166 is, so it seems very likely this should be too. This would benefit from resurveying.