The collision happened at about 3.15pm yesterday (Saturday), in West Lane, Askwith.
It involved a bicycle and a red car, North Yorkshire Police said.
A 23-year-old woman, from Leeds, was taken to hospital following the crash, with serious but not life-threatening injuries.
Police said the red car failed to stop and the force is carrying out enquiries to identify the driver.
They have appealed for witnesses, or anyone with dashcam footage of the area, to get in touch.
Anyone with information is asked to call police on 101, quoting reference number 12180041523.