Hey there, you shouldn't have any troubles finding a parking-slot in Roskilde or Helsingør, just be aware of the stupid parking-guards (I do not know the exact name for them in English).
When you want to park in Copenhagen, just park a bit from where you are going, like if you want to visit Tivoli then park a bit from it, there will be a better chance for you to find a parking-slot, further from the attraction. We do also have parking-houses were you can park for xx days for xxx DKK. My best advice to you would be to search the internet before you are going to the specific place/attraction. While you are here I would suggest you to visit Christania !

bring beers and its easy