The red Mitsubishi 4x4 was found abandoned in the Undercliffe area following the collision in Oxford Road, near Peel Park, at about 3pm on Monday.
A man in his 30s suffered “life-changing” injuries in the crash and remains in hospital.
Three parked vehicles were also damaged along with a telegraph pole.
A West Yorkshire Police spokesman said: “We are appealing to anyone who saw this vehicle being driven in the area immediately prior to or after the collision to contact us.”
Anyone with information is asked to ring PC 5463 Moore at the West Yorkshire force’s roads policing unit on 101, quoting log reference 1047 of September 25.