places to park a car in while visiting Siena

Follow the signs to the stadium (stadio), often symbolized by a soccer ball image. Once you get near the stadium there will be plenty of (pay) parking lots around the stadium. Park and walk to the historical city center, it's just minutes away.

Assuming you are taking the freeway from Florence (Raccordo Autostradale Firenze-Siena), the exit is called SIENA OVEST (Siena West)

We always park at the stadium.

The underground lot at the train station worked well for us. Cost was 2 Euro per day. There's a series of escalators to get you up into the town or, alternatively, you can grab a taxi to the Campo for about 12 Euros.

I agree that parking near the stadium is a good option. I have found parking on the street for free and then took an elevator to the city.

We park at the stadium as well. The entry to the stadium is sort of hidden from the street. I was expecting a bigger road. I had to drive around the block a time or two to find it. GPS was not helpful.

See this.

If you go to Siena on a Wednesday all parking at the stadium is taken up by the weekly market.

It is possible that this is less of a problem with a phone, instead of a dedicated GPS. But approaching Siena with a Garmin some years ago, it proved impossible to get Garmin to navigate to the Stadium lot without (trying) to take us straight through the center of town, and through ZTL's. I finally gave up on the Garmin, and luckily found a street spot, with a steep climb to the city.

For any city, on Google Maps enter 'parking' and the city name. On the resulting map you can click the icons for more info or to get directions to that entity.

Agree with Roberto and others that the "stadio" parking is best. There is a free lot just prior to the stadio, but cars park within inches from another, so better have the "0" deductible if you do.