Donald Trump engaged in a back-and-forth Sunday with the parents of a dead American soldier, extending an argument between the Republican presidential candidate and Muslim immigrants who were largely unknown until they appeared at the Democratic National Convention three days ago.

In an interview on ABC’s “This Week,” Mr. Trump took issue with Khizr Khan, who had questioned whether Mr. Trump had read the U.S. Constitution and said Mr. Trump had “sacrificed nothing.” Mr. Trump responded by citing his hard work and business...