parking in Bergen
I live partly in Bergen and there is no free parking in the centre. But having said that, I have parked my bike in pay zones many times (in fact every time using a bike in Bergen) but never gotten a fine (20 years). I have never seen or heard of any bike that got a fine either. The rule is to park with consideration, near house walls, under trees, any place where it is not obstructing pedestrians or vehicles. Stay away from parkomats, parking houses etc, and parking places for cars. You'll be fine, park where you like but follow the "rules" above!
If you have a car (or bike of course), you could park outside the zone, which is free. It will take you 20 mins to walk into the center. Ask for Sandviken firestation, that's where the zone ends!
Wish you a pleasant stay in Bergen and welcome to our city.