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Hello, I m a solicitor with 20 years experience. I will try to answer this for you.
I take it the access lane is on your land your neighbour has an easement across your land on the lane to his garages (probably on his land). Have I got that right?
If so then as long as you do not block access along the lane with your parking then he can not prevent you parking there. An exception to this would be if there was some sort of binding agreement (eg a restrictive covenant) preventing your use of the lane for parking. Your solicitor would have told you if there was any such covenant when you purchased the land.
Parking on your lane without restricting access is probably something you can do.
Is reasonable access a phrase your neighbour is throwing about? Why does he say you can not park on your own land? If he is alleging this then he should be asked why he thinks this?
You are right he is just being difficult. If you are not preventing the exercise of the right of way you ca do what you want.