If you're not adverse to the idea of driving, I'd suggest driving up from Rome to Siena. That way you can drive through and explore the
Val d'Orcia area on your way up to Siena, it is one of the most beautiful areas in all of Tuscany!
How many days will you have to get from Rome to Bologna? I just want to get a better idea of how many days you could have in Siena to explore the city and the surroundings.
In the meantime, here are some ideas for day trips from Siena:
https://www.discovertuscany.com/siena/what-to-do/day-trips-from-siena.html
You can stay in Siena - a good place to stay at with parking is
Hotel Athena. We've stayed there and it is very central, just inside the walls and close walking distance to Duomo and to Piazza del Campo in Siena. With parking included, it is easy to drive out for day trips and no need to worry about parking in Siena.
Most towns do have their "centers" off limits to cars, known as ZTL areas. As long as you follow P signs for parking lots as soon as you arrive into a town, you won't be driving into those areas on accident.
Take a look through our website and get a better idea of what there is to see and do - even in January everything is open although because of the weather some places are a bit quiet (think of the seaside, being cold it isn't on the top of anybody's plans but it can still be nice to visit as long as you don't expect a dip in the water

).
Let me know if you have more questions, we're happy to help!