Although Copenhagen is generally a quite and peaceful city, it's still a city. But it offers lots of hiding places during the warm season. There are lots of beautiful lush parks in Copenhagen. Some of the best are Kastellet, Frederiksberg Have, and Ørsteds Park.
Kastellet looks like a star-shaped little island surrounded by a moat. It's a perfect spot for a morning (or evening) run because of the path along the rampart. There are also some cosy benches and lawns where you can rest and enjoy the sea breeze. It's very close to the famous Little Mermaid statue. The houses inside Kastellet are used as military barracks and offices.
Frederiksberg Have covers quite a vast area with Frederiksberg Palace towering on a hill. Here you'll find a couple of cute bridges, a Royal teahouse which is essentially a Chinese Pavilion. During summer boat tours are available around the park. Frederiksberg Have is also a great sport for recreational activities and picnics.
Copenhagen's first playground was erected in Ørstedsparken. The park covers 6,5 acres and since it was created after Copenhagen's old fortification structures were decommissioned, part of them remained in the park's landscape. For example, the old moat that now forms the lake which is crossed by a bridge.
Note that some parks in Copenhagen close at night.