Robert F. Smith, the private-equity billionaire who pledged to wipe out the student-loan debt of the class of 2019 at Morehouse College, will cover debt taken on by the students’ parents as well.

The college said Friday Mr. Smith and his family donated $34 million to a Morehouse program that will pay off students’ and parents’ loans.

“With the Student Success Program in place, there is a model for all colleges and universities, starting with Morehouse and HBCUs, to receive gifts from alumni and other supporters that can offset the burden of student loans and give students the freedom to pursue their dreams, the capital to invest in the economic growth of their families and the time that they can give back meaningfully to strengthening their communities,” Mr. Smith said in a statement, using an acronym for historically black colleges and universities.

Mr. Smith, who is CEO of Vista Equity Partners and one of the country’s wealthiest African-Americans, said he would eliminate student debt while speaking at Morehouse’s commencement in Atlanta in May, stunning the college and its students.

Many graduates and their families wondered at the time if the generosity extended to parents who borrowed to pay for their education. Morehouse said Friday that the program would include so-called parent Plus loans, unsubsidized federal loans for parents of undergraduates.