HOUSTON — The couple, farmers from a village in the Mexican state of Veracruz, had traveled more than 1,200 miles to see their 26-year-old daughter, who was dying from an inoperable spinal tumor.

Jose and Ninfa Sanchez, both 48, had applied to cross legally at the Texas border city of Hidalgo under a program called humanitarian parole, designed to allow foreign nationals to come to the United States for emergencies, such as medical crises, court hearings or funerals. It was up to U.S. border officials to decide whether the parents could see their daughter, Maria, before she died.

Humanitarian parole is one of the quirks of U.S. immigration policy. It’s sparingly granted under a system that is largely discretionary.