Parking in Sweden and Norway

I would be inclined to search for lodgings that have some sort of parking available.

Yes, in general cars are allowed free parking until 8 or 9 am, even in large cities like Oslo (if you are in a residential area, not the centrum). Parking in the day usually runs $3/hr.

Be sure to inquire locally. For example, Bergen is a problem with cars. RS Scandinavia Guide: "Parking is difficult and costly - ask your hotelier for tips. Note that all drivers entering Bergen must pay a 25 kr toll." We stayed at a small, cozy hotel in the residential area just up the hill a few streets from downtown - very convenient for walking. Not so convenient for parking. We had to pay something like 25 NOK per day for a spot that was 5 minutes walk (uphill) from the hotel.

Oslo has a toll ring: https://www.oslo.kommune.no/english/politics-and-administration/green-oslo/best-practices/the-toll-ring/#gref
Oslo's tolls are disincentives to reduce emissions/pollution and traffic congestion, and helps pay for transit.

In recent years, significant expansions in the public transport
system, in Oslo, have been financed by a toll ring. The toll ring is
also an important tool for traffic reduction, transition to
zero-emission vehicles, GHG reductions and improved air quality.

You will need to study the Norwegian automated toll collection system: http://www.autopass.no/en/visitors-payment

All drivers, regardless of nationality, have to pay Norwegian road
tolls. The automated toll stations are all equipped with a camera that
reads the vehicle number plates and toll tags. Foreign registered
vehicles, without a toll tag, will be invoiced by Euro Parking
Collection plc (EPC).