RHB is do-able in Folk Week, but only just! Travelling between the two places by bus is really not practical though. We have done it twice (staying at Bay during FW that is!) and found that we really needed the car. However, a weekly parking ticket is not too expensive, and if the worst comes to the worst, the Abbey Car Park is rarely completely full, though you need to be fit to use it regularly.
The Contract Ticket though can be used an any of the Scarborough DC Long Stay Car Parks, which ever place you are in, including Bay.