There are many places to stop, we parked here N 61 46.573 E 29 20.188, near a cafe and toilet, and walked through the forest. We also saw pictures of the observation tower N 61 45.722 E 29 23.076 and went there, but it was closed and abandoned, so we only made few pictures from the drone.
Check the official website for more information about the national park.
Getting there: we parked near the lakes N 61 46.573 E 29 20.188 and near the tower N 61 45.722 E 29 23.076. There is also the Punkaharju railway station. Check here information on public transport.