FWIW I am a private road (read track) owner along with a navigable dyke.
Parking cars is not so much an issue as most plots have space for parking and we have created an area for additional parking. The dyke on the other hand has moored boats and as a riparian land owner (ie I own the land beneath the water), I can allow or not the rights for people to moor their boats if I wish. However, most boats have been moored along this dyke for more than 20 years so established use comes into play and I don’t get any legal powers over that, I could however sell mooring/parking rights if I wanted to. The key conditions for both road and dyke is that they must be open for vehicles and craft to pass and repass, so any car, van or boat that obstructs access I can have removed, although I’ve never tested that one as there has been no need.
Out of interest, do you only have an easement for access? How do your utility services get to your property? If it’s under the road, easements would be required for you to connect to the privately owned services such as water supply, sewage, electricity and gas if you have it. You are in effect connecting to a private supply, your meters being sub meters.
None of that really helps you in terms of what you might like to hear, however if I were in your position, I would create a parking space within the boundary of my property.
