We stayed in a hotel a block off Il Campo and the hotel advised us to park in the free lot behind the Medici Fortezza. Our car was fine there for three days. It's a popular lot, but we found a few free parking spaces in the late afternoon. Earlier in the day it may be full. It's probably as safe a lot as you'll find, unless you pay for a garage somewhere. Just make sure you don't leave anything visible in the car when you leave it.
From the parking lot, it was a 20-minute hike back to the hotel on a hot day, so when it came time to check out of the hotel and take our luggage to the car, we called a taxi. Technically, you can drive to your hotel to pick up your bags upon departure, but the streets were so narrow and crowded with pedestrians when we arrived, I decided not to drive down there again. Taxi to the car is MUCH easier and less stressful.
If I had it to do over again, I would have parked the car in the lot first, and taken a taxi to the hotel. Driving into Old Town Siena should be avoided.