For a very good reason - if you look at the old pictures, you'll often see the pre-1961 sidewalk embedded in the street. The Wall was often build in the middle of streets that had been used as demarcation lines between Soviet and other zones. When the wall came back down, those streets were rebuilt, and so car parking went down exactly along the route of the wall.
You can see this by looking at the ground; modern Berlin marks the route of the wall with a line of bricks in the middle of the pavement. In many of the current pictures, that line runs just between the traffic lanes and the parked cars.
