You can use your TIM card, no need for another one, all TIM cards are pre-activated for using data too.
Data plans are usually monthly or weekly and they are automatically renewed every 30/7 days. You cannot switch them on or off on the days you use them. It's important to turn the offer OFF since otherwise you'll run out of credit and some operators also go below zero €.
I have no idea how much data your parents need every day. I think an option with per KB billing would be better rather than the time billed ones. So TIM is not a good choice.
If you are satisfied with TIM's coverage you can use COOP or Noverca virtual operators.
Otherwise I'd go with Wind.
All information about data in Italy is here:
http://www.prepaidgsm.net/en/italia/data_italy.html