Can I stay in Luca for one month ?

Not sure I understand your question. Are you asking if Lucca is a good base for a month for seeing places in Tuscany? Or are you asking if it's a good place to stay without a car? Or are you asking if you're allowed to stay there for a month? And why wouldn't you be?

Lucca has train and bus service
Of course you can stay there for a month if you wish- without a car- in fact you do not want a car IN Lucca as the old town is completely within the walls where ZTLs don’t allow you to drive
Parking would also be an issue

Do you have an idea of what day trips you might be interested in? Perhaps we can help with that

Thank you for your kind response. I am interested in any suggestions you may offer

Hello swim2535, I just spent a month in Lucca from early September through early October.

I think it is a great little town and while I'm not sure all the restaurants will be open but there are a lot of them. And you will certainly be living with the locals and an absolute minimum of tourists at that point. The town is walkable East to West in about 20 minutes and North to South in about 15 so after a few weeks you'll get to know most of it.

It is pretty well connected by train to the surrounding areas even if the train is the slow regionale train that always seems to take longer than it should.

List of potential day trips:
Florence, Pietrasanta, Pistoia, Barga, Pisa, Castelnuovo di Garfagnana, Viareggio, Forte dei Marmi, La Spezia
Bologna is about 2 hours away which would be my limit but could be doable as a daytrip if you're dedicated.

With a month you can also do weekend trips if you want to pretty much anywhere - even Venice and Rome are in reach at about 4 hours by train. They are too far for a day trip but if you're willing to spend a night or two those places are open as well.

Best Gelato: Ete, Momo (Caniparoli Cioccolateria honorable mention for chocolate)
Best Pizza: Pizzeria da Felice (old school also checina). Itaco
Best Sandwiches: Da Ciacco (also huge salads), Pan di Strada

Get your water from the aqueduct fountains and walk the walls everyday.
https://www.luccagrapevine.com/

If you have any other questions let me know,
=Tod

You can but a larger city and transportation hub like Florence will offer more possibilities for day trips, especially if you don’t have a car.

Would Parma or Luca be a better choice?

Any place works but for a whole month and in winter I’d prefer to stay in a larger city, like Florence or Bologna, which are two important railway hubs which would allow you to easily take day trips to places, even far, since they are served by high speed trains.

PS: Lucca is spelled with two C’s

What do you mean by "better choice"? What are you looking to do? "any suggestions you have to offer" is hard to answer when you've told us absolutely nothing about yourself. You might ger better responses if you offered up more information.

I have been given some excellent suggestions. Should I fly to Rome or Milano connect to Pisa or train from either destination?