Can parent of child in dom rep sue absent parent in US for child support?

The short answer is yes. Where and how depends on a lot of issues including: 1) is there a valid decree in the Dominican Republic and 2) where in the U.S. does the absent parent live? Obviously, it is more difficult to enforce a decision across international boundaries than across state boundaries, and there are additional risks for the custodial parent. But, in theory, most states recognize a common law obligation for the absent parent to provide child support and courts are open for foreign citizens and foreign residents to pursue claims against individuals residing in the state.