Hi Kate,
Traffic is strictly limited inside the city walls unless you have special authorization from your hotel to park inside the city walls.
I would say parking near the city walls has become quite impossible. Most of the free lots such as near the
fortezza are taken up by locals early in the day throughout the day and the only parking available has been metered. It is around €1.50 an hour or €47 a day!
There is a free parkling lot next to the fortezza and a pay lot on the other side of the fortezza next to the La Lizza park; there is also a pay lot nearby at the nearby soccer stadium.
However these pay lots are unavailable on market Wednesdays and soccer Sundays. Parking along the street next to the fortezza is free, although if you park down some of the sidestreets it is metered. Check first.
There are also pay parking garages/lots at Porta Tufi (signposted "Il Campo Parking"). It is a 15-20 minute walk to the
Piazza del Campo. And another pay lot at Porta San Marco, as well as metered parking on the street leading into Porta Romana.
There are parking garages by Porta Ovile (metered), depending upon where you're going inside the walls. Either way check with your hotel or landlord to see what your parking options may be.