Most people who come to hike Pico Duarte [1] stay in Jarabacoa [2] or Constanza [3] hotels—that is, until they are on the actual hike. There are about a dozen campgrounds along the way providing very basic shelter. These cabins are basically an empty wooden box and do not have beds (you sleep on the floor). It is not possible to reserve a cabin and they are not tended to by the parks on a regular basis. There are kitchenettes outside of the cabins to cook your meals.
Links:
[1] http://www.moon.com/destinations/dominican-republic/el-cibao/parques-nacionales-bermudez-y-ramirez/pico-duarte
[2] http://www.moon.com/destinations/dominican-republic/el-cibao/jarabacoa
[3] http://www.moon.com/destinations/dominican-republic/el-cibao/constanza