Joseline,
The "old phones" you have there MUST BE unlocked quad-band GSM models or they WON'T work in Italy! It's unlikely that any Sprint phones would be able to connect to GSM networks.
One other option you might consider would be a "travel phone" from a firm such as Call In Europe, Roam Simple or Cellular Abroad. When using these networks, anyone calling you from the U.S. would have to dial a European number, so long distance charges would apply. However, if you were using an Italian SIM card and phone plan, the same situation would exist. Would your parents have any difficulty dialing the numbers for international calls?
The easiest and simplest method would be to use roaming with AT&T or T-Mobile, however that's not necessarily the cheapest method. The one advantage is that your parents would only have to dial your normal phone number (therefore no long distance or complicated dialing sequences for them to deal with), and the network will find you wherever you are.
I believe AT&T has an "international roaming plan" that reduces roaming charges somewhat. This can be activated just prior to departure and then de-activated when you get back.
If your phone contract is due for renewal, perhaps switching to one of the GSM networks would be an advantage?
You could also buy an unlocked quad-band GSM handset off E-Bay and then buy an Italian SIM when you arrive. If you do buy a phone make sure the Charger supplied with it is designed for world operation! A Plug Adaptor will still be needed though.
I always use roaming with my home network when I travel, and it's worked well so far.
Good luck!