Road bikes do fine. Being city streets there are some potholes and paving joints to watch for, but most of it should be ok. There used to be some really rough patches in Queens and Brooklyn, but they repaved those parts a couple of years ago.
Be especially careful on the bridges - a couple of them the big finger-jointed expansion joints that can eat skinny tires.

Past rides I've seen tandems and bent tandems, trikes, handbikes and even a Pennyfarthing. A tandem shouldn't be a problem. Once you're into Central Park, the crowd tends to thin out a bit.