I have to admit that I've never been to Tyresta Nationalpark. But you should know that at this type of place you are NOT allowed to wild camp. The human impact is kept at a minimum. You may ONLY camp at prepared sites and may ONLY make fire at prepared places. Hardly wild camping...
Here are the rules.
But you should be able to camp at night at the prepared sites and go exploring in the daytime in the park.
I would expect it to be a little crowded, however...
If you would like to truly wild camp, on your own, then you should go anywhere except to a National park. In Sweden you may wildcamp just about anwhere for a night, as long as you don't in any way disturb or destroy, taking advantage of
the right of public access. Places near big cities are probably not the best. Hop on a train or a bus for an hour or two and try to find some isolated spot inland where you can go exploring without meeting anybody at all. That takes care of the "don't disturb" bit...
Or be a little less wild and take advantage of some prepared walking route with shelters, like Sörmlandsleden. Not too far from Stockholm.
http://www.sormlandsleden.se/etappskyltar/Eng-hemsida.html
Sörmlandsleden partially includes Tyresta Nationalpark. So you could start there and check it out, and walk down the trail and see where you end up... 8)
A.