I'm tempted to change it to bridleway as it seems to be used only by horses which aren't allowed to take the bridge, but as far as I know horse riding isn't allowed in the national park at all, so this would also be questionable. Besides, you could also blame the routing engine if it routes you over a ford of 70 meters instead of the bridge right next to it (don't know if depth=?/surface=mud would change anything), nevertheless I'm also not convinced by the current tagging.