If a landowner wants vehicular access, he can apply for a vehicle crossover to provide access to a drive or other off-road land, and the council will normally install a lowered kerb to allow easy access for vehicles. However the fact that this has been done does not mean that the part of the pavement in question is not still legally the pavement. The fact that it might look like a lay-by is irrelevant, you car was parked on the pavement.
To pursue an appeal we'd have to find something else to argue.