By Agence France-Presse
"The male driver of the car was detained by officers at the scene," the police statement said. "A number of pedestrians have been injured."
Police did not say if they suspected terrorism, calling it only a "collision".
Armed police immediately surrounded the silver car after it crashed at 7.37am (06:37 GMT), pointing guns at the driver as he was removed from the vehicle, according to footage posted on Twitter.
Later images showed police holding the man, dressed in jeans and a black puffer jacket, in handcuffs as roads around parliament were sealed off.
Westminster was the scene of a terror attack last year, when Khalid Masood, a 52-year-old British convert to Islam, drove a car at pedestrians on a bridge over the River Thames, before fatally stabbing a policeman on guard outside parliament.
The attack left five people dead and around 50 injured, and only ended when police shot Masood dead.