Artem Marchuk needed to escape Ukraine or die. He didn’t see any other options.

He and his wife and children had been living in Bakhmut, the site of the war’s deadliest battle. Even when they made it out of the city, nothing in Ukraine felt safe.

“My kids were very hungry,” Artem’s wife, Yana, said in an interview from the family’s home in Baltimore, where the U.S. government resettled them in 2022. “There was darkness everywhere.”

The Marchuks are among more than a million people whom the Biden administration has allowed into the United States over the past three years under an authority called humanitarian parole, which allows people without visas to live and work in the United States temporarily. Parole has been extended to Ukrainians, Afghans and thousands of people south of the U.S.-Mexico border fleeing poverty and war.

Now the program is at the heart of a battle in Congress over legislation that would unlock billions of dollars in military aid for some of President Biden’s top foreign policy priorities, such as Ukraine and Israel.

Republicans want to see a severe crackdown on immigration in exchange for their votes to approve the military aid — and restricting the number of people granted parole is one of their demands.

For Mr. Marchuk, the fact that a program that saved his family has become a bargaining chip on Capitol Hill feels wrong. Although the latest version of the deal would mostly spare Ukrainians seeking parole, he feels a deep sense of solidarity with other people — regardless of their nationality — who may be left behind if Congress imposes limits on the program.