When we're doing work on city streets like you're describing, we try to either block off the area the night before to prevent parking or call the local police department and have them do it for us. Usually they'll either post the area if given notice enough or they will actually contact the vehicle owners by way of their license plate numbers to have them moved. And as far as street closures, if given 24 hours notice to all the emergency services(police, fire, and ambulance) we are able to close down the street and reroute traffic, assuming it's not a major thoroughfare. If so, then it's flaggers, signs, and cones.
Even then there are still idiots, though. We have had several people drive through the drop zone, even after being flagged through on the detour route. I was dropping about a 40 foot limb onto a street, pulling with my skid steer, and a guy stopped under the branch as it was hinging over. Thankfully for him, he looked up to see what the skid steer was hooked to or he would have been a dead man. He flew out of the area as fast as his little S-10 would take him.