Cuba more than doubled some doctors’ salaries last year, to about $70 a month. But faced with such small salaries at home, many doctors accept a posting overseas to make extra money.

Still, they earn only a fraction of what the host country pays Cuba for their work.

Dr. Mara Martínez, a dentist who is Dr. Sánchez’s fiancée, said she was a staunch supporter of the Cuban revolution but became disillusioned when she arrived in Venezuela to find that she had to work six days a week and sleep three to a room for a salary of $210 a month. Venezuela, she was told by her supervisors, was paying $7,000 per month for her services.

“It’s modern-day slavery,” said Dr. Martínez, 25, who left Venezuela with Dr. Sánchez.

Many of the doctors in Venezuela are stationed in cramped quarters without air conditioning, overwhelmed by the country’s soaring inflation and deteriorating economy. “It was worse than Cuba,” said Dr. Dailanis Barbara Martínez Peralta, a general practitioner who after leaving Venezuela waited seven months in Colombia to come to the United States.

More than 7,000 Cubans have been approved for residency since the program began almost a decade ago, according to United States Citizenship and Immigration Services.

According to Homeland Security statistics, 1,663 Cuban medical professionals posted overseas were accepted to enter the United States in the 2015 fiscal year, a 32 percent increase from the year before. The number of doctors admitted to the program has more than tripled since 2011, when 386 people were approved.

Cuban officials have repeatedly assailed the medical parole program as a “reprehensible practice” aimed at “stealing” Cuban talent. A State Department official, who spoke on the condition of anonymity, said the United States “does not recruit Cuban medical professionals,” but simply gives them a voluntary route to residency.

In the spring, American approvals of doctors seeking asylum through Colombia slowed considerably, in what analysts took to be a gesture of good will as the United States prepared to open an embassy in Havana. The State Department denied any connection between the delays and warming relations, and said there were no “immediate plans” to scrap the parole program.