Nope, the OP owns the WHOLE drive. However the neighbours have the right to use it for access reasons only. This isn't a case of shared ownership, or there being a theoretical line down the middle with the OP "owning" one half and the neighbours each owning the part of the half that is outside their own property.
The countryside should be shared (as long as you’re not impacting anyone else) there was no no parking sign, no private road sign, nothing. I know I’ve gone off topic but I’m so sick of this what’s mine is mine attitude (unless they’re rude and disrespectful of course then by all means exercise your rights). But I just think if people were a little kinder to each other the world would be a much better place
Firstly, how do you know you weren't impacting the farmer in any way - others up thread have given reasons why that area may have needed to be kept clear. And what's mine is mine? Too fecking right it is - they've paid for it, they maintain it, same as you pay for and maintain anything that you own. If there's a gravelly/grassy patch beside the road alongside a field, there's likely a very good reason why that hasn't been overgrown (like it's in constant use). If there was no solid growing hedge or hard walling there, it could easily be access to the field and no, they don't always have gates as such, just places where they can temporarily move the wire/fencing.
And if it was truly a public lake, there would be other forms of access that didn't mean parking on someone's private land. They may not have been convenient for a small child due to walking distance, but they would be there.
In my county there's a very beautiful stretch of river running through a private estate that has a public right of way along the bank. Despite signs saying "private land, keep to the path" you get hoards of people setting up for the day on either side of the clearly defined path including on the grazing land. Obviously the owners can't have animals grazing there during the summer months for the safety of both the animals and the public, but the clean up job every autumn to make it safe for the animals is immense - plastic, food wrappers, disposable barbecues, bit of metal, discarded blankets, broken glass....
Plus the river is not a public waterway, it is privately owned. You get dozens of people swimming in it on a hot day, and it isn't safe for that due to the rubbish that other people have dumped in it. Local kids jump off the bridge, and there's a couple of rusty bikes that have been dumped off that so not safe. Then when these members of the public get hurt, they try to sue the land owner!