FOR NOW they can only take action against the driver. When the new laws are passed this year parking enforcement companies that are members of the "trade body" (BPA) will be able to take action against the *owner* of the car, irrespective of who was driving it.
There's a petition against this huge extension of their powers on the government petition website but it seems to be under most peoples' radar. The powers are conferred under Clause 56 of the Protection of Freedoms Bill (should read Erosion of....). You can read about it and access the petition at: