To your phone question, you may wish to check who your parents' current mobile phone service provider is, and if they operate here in the US. I mention this because, when we visit the UK, we take along a couple of T-Mobile phones that we no longer use in the US. I get free SIM cards from T-Mobile in the UK, and I add minutes at grocery stores, supermarkets, fuel stations, (or a T-Mobile store).
Aside from this, there are a couple of options, but the most covenient will probably be to purchase a 'pay as you go' phone(s), either at one of the US service providers stores, or at a store like WalMart. They can then purchase minutes for their use. When I did this for a neighbor a few years ago, I bought the phone and minutes at a WalMart store, and they gave me a lot of extra 'free' minutes. It was surprisingly inexpensive.
The alternative to the above is sign up with a US carrier for a service plan; The phone will be relatively inexpensive, but the carriers usually require a 2-year plan. They'll also ask for a US address, and possibly a US social security number.
Whichever option they choose, you should check out the coverage maps on each carrier's web site, and look for coverage in the areas they'll be traveling. There's also an issue if they expect to make calls back to.Italy; This can be very expensive, and usually requires activation of the international feature by the carrier.