Playgrounds in Oslo
There are 32 public playgrounds in Oslo. You can find an overview of these on the pages of the City of Oslo's pages.
I've made an attempt to make a map of these playgrounds, but Google Maps had problems locating them, so these locations are very approximative!
Vis Lekeparker Playgrounds Oslo i et større kart
If there's no playground where you live, there are other possibilities.
Day cares (barnehager)
it's common practice that you can use day cares' outdoor areas outside of the opening hours, which would mean approximately before 07.30, after 17 and in the weekends. If you don't feel comfortable doing it, feel free to ask the people working there. Here is an overview of the day cares in Oslo.
Open day cares
Many open day cares are open for kids up to 5-6 years. Read more about these here.
Children's parks (Barneparker)
Barnepark is a good alternative for parents who don't want their kids full time in regular day care. These parks are open 4 hours a day, and they have grown-ups looking after and playing with your kids. You have to pay for this, but you keep your "kontantstøtte" (if you have it). The number of barneparker has been drastically reduced over the last years, and today there are only two left in Oslo: Sofienbergparken and Kløverveien.