We like parking at these parking garages that are near the old town. They also have direct freeway access, so you don't have to drive around lot of small crowded streets.
Bahnhof Parking Bern (Parkterrasse 14, 3012 Bern). If you are staying more than 2 hours, don't park in the "Kurzparking" (aka short term parking). which is more expensive. bahnhofparking.ch