Can you park our rental car for a few days near siena it, such as at the rail station?
I'd probably park in one of the easy to use and easy to find garages. For us the station one was just too far away.
The garage we used, can't remember the name just now but it was well labeled as we drove up the hill, was very secure, well lit, and close to the Cathedral (you've got to see this Duomo).
Steve has good advice about giving the car back instead of paying for parking.
But maybe they are staying in Siena several days and using it as a base to explore the area by car---in which case they need to keep the car.
Maybe consider staying jut outside the city at a place that offers parking. We spent one night at a lovely agriturismo just up the Chiant Highway, about 8 miles north of Siena. It offered free parking, and numerous swimming pools, including a "river" with a current and one with a waterfall. It made for a grea overnigh stop during the heat of late June.
Before you jump on the two-car idea, check out the cost of one longer-term rental vice two short-term ones. There might be a big price differential.
Also, you'll need to figure out the time you'll most likely actually be parked, rather than the max twenty-four hour rate.
Then figure the time spent checking in one car and checking out another, plus digging around under the seats to make sure you have all of your stuff. Your time costs something when you're spending a wad of bucks each day.
I've never spent the night in Siena, but I'm sure I've seen cheaper rates (maybe even free) in lots a bit further away from the center.
If you are interested in staying very close to Siena, but not having to worry about a place to park your car, you might want to consider Agriturismo Marciano. We will be staying there in September, so can't give you a first hand recommendation, but it is highly recommended on Trip Advisor. It appears to be about a 5 minute drive to Siena.