Permitting a 'carriage' to cross or park on the pavement, verge or footpath is as offence under s72 of the Highways Act 1835. It is the same act that prevents a bike being ridden on the pavement - carriages include many types of vehicle.
This is why your council is saying you are committing an offence by driving across it.
The pavement etc. is part of the public highway, so the way around it is to formally apply and be granted consent to access your land from the public highway. Then it would cease to become an offence for you to cross it but youstill couldn't park on it.
The parking on the pavement etc. is different. There is no general right to park on the road (the normally metalled highway), even if there are no parking (local) restrictions in force. However it has become accepted that vehicles can do it unless they are causing an obstruction, when it is typically dealt with by the police, but some local authorities have taken on local powers to deal with it. By the same token, parking on the pavement can be classified as an obstruction to pedestrians wanting to use it, so that is also often dealt with by the police.
By having a 'proper' access, which will include a dropped kerb, you have more 'rights' to expect that vehicles will not park on the road to prevent your access, since to do so the offending vehicles would be causing you an obstruction.