There's a well-word desire line in the grass (Well, the dirt) between the Holyroodhouse car park booth and the park exit by the old Elsie Inglis. I've been known to take that as a shortcut on the bike, but never been sure what the legal status is. Given I go past the front of the ranger station and nobody ever yelled at me yet, it can't be that serious an infraction. Also, it's either dirt or your basic park grass that I'm riding on, I wouldn't dream of taking a bike up onto the hill proper.
If I was stopped and challenged and was told "your'e ruining the grass" I was going to point out that it's the 1,000s of people who walk that way as that's the way they want to go and they don't want an 800m detour that are "runing the grass" and I'm just following slowly in their footsteps.