It’s depending on the situation. If parking is forbidden for none-residents, amenity=parking should have the access=private, if it’s forbidden as well to use the service road, then the service-road should get an access=private as well. No router will check, whether any road will cross a private “area”.

In the end: The access of the parking will influence whether you see the parking in the list of your POI and the access on the highway will influence, whether the router can use the road.

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