Yasuní National Park is a wildlife haven like no other.

In eastern Ecuador, where the air carries an earthy scent of vanilla, towering kapok trees rise in a dense, unbroken canopy, their branches heavy with vines and epiphytes. Branches flash with colorful birds, while a vast river teems with pink dolphins, giant river otters and hundreds of species of fish.

This richness doesn't stop at the park's boundaries — and neither do the pressures on it. Oil deposits have drawn companies into the region, and the roads built to reach them have opened land that was once remote. Settlers followed those roads, clearing forest for farmland in the fertile soil left behind.…