I know its been quite a while since the question has been asked, but if someone still finds this useful, here's my two cents.
I was in Denmark a few months ago, and parking in central Copenhagen can be really expensive. However, just outside the center it is not too difficult to find a cheap or if you are really lucky, a safe and free place to park your car and continue on public transport. Visit Copenhagen has some very valuable information about traveling and parking in Copenhagen.
I have not used it my self but a friend who was there for Christmas has found a mobile app which can be really useful in finding parks and public transport transits from there. Here's the link. Copenhagen parking app