You need to get a permit or be accompanied by a park ranger. At the entrance points there are park ranger stations. There, I suggest you go the day before and arrange for a guided visit.
Parque del Este is an enormous area. There is lots to see. Depending at what point you enter. There are many caves to visit, eco trails (for instance there is one near the town of Boca de Yuma). There are also specialists that offer guided visits to observe flora and fauna.
