Parents, university coaches and a college-admissions counselor were among dozens charged March 12 in a sweeping criminal conspiracy that sought to help applicants win admission to elite schools including Yale, Stanford, UCLA and Georgetown.

Wealthy parents—including celebrities, a top mergers attorney and a venture-capital CEO—are alleged to have paid bribes to get their kids into school, giving cash to test-takers to help students cheat on entrance exams and paying coaches to designate applicants as athletic recruits. Parents paid from $100,000 to $6.5 million in bribes, with most payments around $200,000, according to prosecutors.

“The parents are a catalog of wealth...