One of my daughter's absolute favorite places in Copenhagen is the playground called "Byoasen" located in Nørrebro.
Byoasen is not an actual playground, but it's a green area in the middle of the city where kids can hang out with animals and help take care of them eg. by feeding them.
Goats, rabbits, hence, mice and guinea pigs are animals that can be found here.
Entrance is completely free and everything is very down to earth, and as long as the kids treat the animals carefully and with respect they are even allowed to hold them (except for the rabbits - well and the goats of course).
We can spend hours here especially on days with nice weather and afterwards we usually go grab a coffee in one of the many cafes located just around the corner in Nørrebro.
For more information visit the website of Byoasen.