The incident happened at the 2017 Wakefield Classic And American Car Show in the city centre yesterday.
Police reported that officers and paramedics had been called to Westmorland Street during the afternoon following reports of a collision.
In an online update, West Yorkshire Police's Wakefield Central team said: "In a freak accident, it appears that one classic car lurched forward at speed, pushing another car backwards into members of the public.
"Two people were taken to hospital, but thankfully there appears to be no serious injuries."
A message on the event's Facebook page said: "Great show today, all fantastic cars and thousands of people in the city centre spending money back in our city.
"Minor incident at the end which was unfortunate but the people concerned are fine."