Personally, I think you will really have your work cut out for you just managing to do Venice and Rome in 3-4 days, and can't imagine how you would also squeeze Paris in as well. I would save that for a another trip.

Actually, I would consider doing just Rome OR Venice in such a short period. I would start by getting a good guidebook, reading it, and thinking about the things in Italy that will most interest you and your parents, and going from there. I think Venice is particularly well suited to a weekend first trip to Italy but, depending on your interests, you may decide otherwise.

Are you near an airport that has an airline providing discount flights to Italy? You could consider that or train.

WHerever you go, one of the best ways to keep costs under control is to book ahead -- plane, train, hotels, whatever. Read the hotel reviews on the this website for the places you decide to visit and look for well-reviewed ones in your budget. Then check for hotel websites and see what kind of prices you are offered there. I have had good experience booking directly through hotel websites.