Maggie,
I did the same as you, traveling with a leased car and stopped for a few days in CT. I was there in late October and could have probably parked in the small lot above Vernazza that time of year. I wouldn't even try in August.
I anticipated not being able to park, so took the local train in from La Spezia.
Rick describes some parking around the train station. It was packed and didn't look very secure (broken glass, loiterers, etc.) so I drove back in towards town and found a free place to park accross the street from the Navy base. It was a good 10+ blocks to the train station.
I found parking by asking two people. The first pointed me in the general direction. I found parking but wasn't sure it was okay to leave a car there for several days so I asked a sailor. He pointed me down the street. It wasn't a parking lot, just parking on the side of a fairly big street. It was free and the area seemed much safer than around the train station. The car was fine when I got back.