Here's a novel idea... have you tried talking to the neighbour and telling him that he may no longer use the driveway you own?
Kind of drastic..........
Maybe I missed something, but...........Did you at least try ringing his doorbell, introducing yourself as the new neighbor and telling him you now own the house and....."please don't use the driveway anymore"?
You can always leave him a note ( in his mailbox) if you can't catch him at home.
If it continues, and you can't track him down after talking to other neighbors......then call the police - again............
If you are not sure who's vehicle it is you could still talk to the neighbors. You can knock on some doors, & introduce yourself, then you can explain your problem & see if they know whos' vehicle it is. They might know & guide you to the right home. When you do find the right person, you can explain that you have purchased the home now & they can no longer park their vehicle there.
If you do not want to ask the neighbors, then you could leave a note on the vehicle. Before the home was purchased, I'm sure there was a lot of cars there for showings. They may not realize that the home is purchsed now.
It's evident that someone is hoping you'll go away......don't let it happen!