we did the same.... the brilliant old manager let people park on the grass as a solution. I had a huge clean up when I made and enforced rules. Looking to now add a company called Parking Snap where guest register in a database and are auto denied after 2 nights.
Our specific issue stems from the city only requiring 1.5 parking spaces per apt so there really isn't enough parking for each lease holder to have a car.....
We do parking permits as well, which will cut down on non occupants parking. Since the city limited your parking though maybe you can give incentives for not bringing a car-free bus passes for example.
At a property I used to manage there was a monthly fee for parking so maybe that would cut down the number of people that bring cars. Just a thought.
TRANSPARENCY! Explain upfront the parking situations and put in place a parking addendum with permits. You might lose some leases because your property is just not a good fit for the prospect but it's way better than dealing with the dissatisfied resident once they have moved in.
we also use parking passes that adhere to the inside windshield. For non authorized vehicles I have a neon pink parking violation sticker that warns this vehicle is tagged to be towed. It is amazing how fast unauthorized vehicles move without you having to talk to anyone about it.