Crews from Todmorden and Mytholmroyd were called to Burnley Road along with police and paramedics at around noon today.
A white Vauxhall Corsa, a white Citroen and a red 4x4 had crashed near the junction with Lennox Road.
A West Yorkshire Fire Service spokesman said they found a passenger was trapped in one of the cars involved.
He said the person was freed by firefighters before being taken to hospital by ambulance.
The road has since reopened to traffic, but police are now trying to trace the 4x4 after it failed to stop at the scene.
A spokeswoman said the vehicle was believed to be a Land Rover or Range Rover.
Witnesses to the collision or anyone who can identify the vehicle are asked to call 101, quoting log 685 of 28/09.