King's Garden or Kongens Have is one of the locals' favorite parks, especially for a picnic. Located in central Copenhagen, right next to the Rosenborg Castle (which is also worth a visit), it's surrounded by beautiful historic Danish architecture.
During the late-spring and summer months, the park is full of beautiful flowerbeds and is home to a lush rose garden. There are several sculptures around the park including the famous statue of Hans Christian Andersen.
During the Copenhagen Jazz Festival, you can also hear some great music while sitting on the lawn and sipping your drink. And yes, drinking outside is allowed in Copenhagen, so you can bring your favourite drink.
The park isn't that big so be prepared that you probably won't be all alone. But this doesn't have to be necessarily a bad thing, you can think of it as an opportunity to feel like a local.