Hello,
I did some searches to answer your question and I’d say that you can go both by train and by bus from Siena to Florence. The choice is up to you!
Train
As you said the rail station of Siena is outside the center so you won’t have any problems to park your car since there is a big parking lot underground.
Check on
Trenitalia the train schedulesfor direct trains from Siena to Santa Maria Novella (central station of Florence). You’ll notice some trips are longer than others, it depends on the train you take and the time of its departure.
Bus
There is also a convenient bus line from Siena which goes to Florence many times a day. Here is a
time schedule for the buses. The first stop in Siena is in Piazza S. Domenico close to the stadium; the bus also arrives to the center of Florence right outside the SMN train station.
Remember to buy the tickets before getting on the bus, you’ll find it in some nearby bars or stationery stores.
In the end since you are in Palazzuolo and you have a car, I would suggest you make your way to and also discover the wonderful Arezzo! It is quite close and another gem not usually on people’s itineraries.
Enjoy your journey!