You can park in any blue zone parking in Bern for 1 hour free of charge, use the normal blue disk.
You can also buy parking for 4 hours (Costs Fr 8,--) or 24 hours (Costs Fr 15,--) at any (red coloured) Bern Mobil bus & tram ticket machine.
Display the purchased parking ticket carefully behind the windscreen. The parking period starts when the ticket is printed out.
If you buy the 24 hour ticket, you could drive to other parts of Bern and use the same ticket.
http://www.bern.ch/leben_in_bern/woh...eren_quartiere
Google translation,
4- und 24- hour parking permits
For visitors to the city of Bern permits are available for 4 hrs and 24 hours. These can be bought for CHF 8.00 or CHF 15.00 at the ATM of any Bernmobil ticket machine, or at the main police stations east and west, the Park and Ride at Neufeld, and the Police Inspectorate of the city of Bern. These authorizations are valid in all zones of the city parking tickets Bern.