My weak Slovak makes it hard for me to research facts, so I tend to rely on my wife; but I understand that unfortunately the parkland is not "left on its own".
The state fells a lot of timber in the parks. They use the borers as an excuse, but cut down more than they need to control them.
There's yet another sad story for forests among the negative externalities from the transition from socialism. There were and still are large areas of land divided up and owned by general public, which has a legacy today, see [1].
One opportunist where I live went and bought as many of these as he could, often cheaply off old people and made bank by letting people come and fell the forest on them. You can see it all around here - big swathes of bare earth cut through forested hills/mountains with no regard to sustainability. It really upsets me.
[1] http://4liberty.eu/land-consolidation-in-slovakia-chance-for...