The pair, thought to be in their 60s, were hit on Finsbury Terrace in Swansea, South Wales Police said.
The woman was flown by air ambulance to hospital and is conscious, the force said without identifying them.
A blue Peugeot has been loaded into a container and removed from the scene.
South Wales Police said the incident was being investigated.
"Initial reports are that a vehicle has rolled down the street trapping a male and female underneath the vehicle," the BBC quoted Inspector Jason Herbert as saying.
"We released two casualties from underneath the vehicle," she said.