As an alternative to a hotel, I've had relatives stay in people's homes that they rent out from craigslist. This is becoming more common, and is usually much, much cheaper than a hotel. The plusses, besides cost, are you get some ameneties of home like a kitchen, coffee machine. The downside is no personal service.
Go to the "vacation rentals" link at craigslist.
I'd say do not miss a walk over the Brooklyn Bridge or a visit to the Met Museum of Art -- have a drink in its rooftop garden among the Rodins and you get a great view of Central Park from there. There are so many great things to do. It really depends on you and your parents' interests. I would avoid the tourist trap of the tour busses and would choose a cruise around the city instead.
Do not rent a car. Impractical and expensive. Taxis are plentiful.
Have fun.