Police are seeking witnesses to the collision on Saturday lunchtime, which left the woman with serious injuries.
The woman, who is from West Yorkshire, was taken to Leeds General Infirmary where she was said to be in a serious but stable condition.
The white Toyota Auris was in the parking bays opposite Lloyds Bank in Cambridge Crescent when it collided with three vehicles in the bays and the elderly woman.
The driver, a 90-year-old woman also from West Yorkshire, suffered injuries herself and was taken to Harrogate Hospital for treatment.
A police spokesman said the incident took place at around 12.50pm and the road was closed until 4.30pm to allow officers to investigate.
Any witnesses who have not already spoken to police are asked to call TC914 Dave Lever, of the Harrogate Road Policing Group, by dialling 101, selecting option 2 and asking for him by name.
Please quote reference 12170003631 when passing on information.