Sienna and hillside towns in Toscany

One consideration is that the train station is outside the town, so that's an extra bus or taxi every time you leave town. I would really nail down your research so you know exactly what you want to see, then determine where is best to stay. The answer really is "it depends." How many days do you have? I find the best strategy is to stay in a town/small city a few days, and them moveto a village for a few days for exploring more rural areas by car.

This might depend on which towns you want to visit.

Train service is not great for smaller towns in Tuscany, though you can reach a few directly from Florence (but not the ones people generally recommend most). You can get from Florence to Siena by train, but the train station in Siena is below town, so generally the advice here is to take the bus. There is also bus service from Siena to towns to the west like San Gimignano and Volterra, and south towards Montalcino, though I don't know what the timetables are like and I imagine they take a while since they are local buses.

Once you are out of Florence, driving in Tuscany I found to be pretty easy when we went in October. We stayed outside Montalcino for a week with a car, and visited Siena, Orvieto, and Montepulciano from there. Depending on your arrival/departure times, you can also rent cars from Siena or Chiusi or other places to avoid driving south from Florence.

There are tour operators that will do day trips with transportation included from Siena or even Florence if you want to go that route. We did a winery tour day with Ciao Fabry Tours that was fantastic, 2 wineries and a sheep farm for lunch. Fabrizio also does a lot of other trip options and even cooking classes - I wish we had booked more with him than just the one day! https://www.ciaofabrytours.com/

Trip Advisor shows a number of day trip and tour providers based in Siena, though most of them tend to originate in Florence.
Siena is a great base in general - particularly so after the day trippers depart around 5 PM.
If you're willing to reconsider renting a car, know that there are a number of rental car offices in the vicinity of the train station and that there's cheap parking in the underground lot fronting the station itself. The lot is located right on the SS2 road which makes it easy and relatively hassle free getting in and out of the town.
From the parking lot there's a series of people-mover escalators located adjacent to a shopping center that will get you up into the city, or alternatively you can just book a taxi for door to door service to your hotel ... cost is less than 15 euros.
Personal opinion only, but we thought that driving in rural Tuscany and visiting the hill towns on our own by car was an absolute pleasure.
April/May is a great time to visit the area.

We had a great long weekend in Siena a few years ago. We took the train and cabbed it to our delightful BnB. A highlight was the one day winery plus tour we did with Tuscan Escapes. They are Siena based. Small 8 people group.
I would love to do more tours with them. I think you can do Siena and area if you are like us & do not want to drive.
Enjoy

Could we do this without renting a car?

Not easily. We stayed at a lovely agriturismo just outside Greve a few years ago. We did have a car and used it to travel to several Tuscan hill towns plus Siena. We took the bus into Florence for the day, leaving the car in Greve.

The Tuscan area, especially Chianti is not a big area and there's no train service to speak of. Bus service is spotty and not a good option to move from one town to another. If you don't want to rent a car, you could take guided day trips from either Siena or Florence but that would be an expensive option.

Driving in Tuscany is a delight and the scenery is lovely. Just follow the rules of the road and assume anywhere within the walls is ZTL, but we found all ZTLs well marked and easily avoided.

If I decide to rent a car in Sienna and drive around to visit the small hillside towns, how is the parking situation in/around this small towns?

Parking in and around the hilltop villages will be located outside the walls of the old city. Anything inside the walls will be restricted to the locals and will carry a hefty fine if you stray into one of the ZTL zones.
We had no trouble on our most recent visit locating parking in Tuscany by using one of the mapping apps (Waze, Google Maps, etc.) to locate nearby parking, sort out the cost involved, and preview the walk from there to get into the center of the town. Not a big deal, but it does require some advance research.

To the OP, thank you so much for posting this thread. We are staying in Sienna in May for 3 nights, and we plan on renting a car to tour Tuscany and visit the hillside towns. So everyone's responses are helpful to me, too. Especially Robert's responses. Thank you!