In a stunning reversal, a federal appeals court overturned the fraud and conspiracy convictions on Wednesday of two parents found guilty of participating in a far-reaching bribery scheme, known as Operation Varsity Blues, which ensnared dozens of wealthy parents who falsified their children’s credentials to gain admission to prestigious universities across the country.

A three-judge panel of the U.S. Court of Appeals for the First Circuit in Massachusetts found that the lower court had made crucial missteps in the trial of Gamal Abdelaziz, a former casino executive, and John Wilson, a private equity financier. The court, however, upheld Mr. Wilson’s conviction on tax fraud.

The appeals court made its decision largely on two technical legal grounds.

First, it ruled that the lower court judge wrongly instructed the jury that admissions slots constituted property. “We do not say the defendants’ conduct is at all desirable,” the decision said. But the appellate judges faulted the government for being too broad in its argument, to the point where “embellishments in a kindergarten application could constitute property fraud proscribed by federal law.”

The court also found that the government had failed to prove that the two men agreed to engage in a conspiracy with other parents, who were, like them, the clients of William Singer, known as Rick, a college admissions consultant to the rich, the mastermind of the admissions scheme.

The conspiracy charges allowed the government to introduce evidence related to other parents’ wrongdoing, creating the risk of bias against the defendants, the judges said in a 156-page decision.

A spokeswoman for the U.S. attorney’s office for Massachusetts said on Wednesday that the office was “reviewing the opinion issued by the First Circuit Court of Appeals and assessing next steps.”

The victory in the appellate court was striking because Mr. Wilson and Mr. Abdelaziz were the first to take their chances in front of a jury. Dozens of other wealthy parents, including some celebrities, pleaded guilty, making it seem as if the prosecutions were ironclad. The investigation became a symbol of how wealthy, prestige-obsessed parents had turned elite universities into brand-name commodities.

“Almost everybody pleaded guilty, so the government’s legal theories weren’t really tested until this case was decided,” Joshua Sharp, the lawyer who argued the case for Mr. Abdelaziz, said on Wednesday.

Noel Francisco, Mr. Wilson’s lawyer, said the appellate court’s decision showed that his case was “fundamentally different from others in the broader Varsity Blues scandal.”

While Mr. Abdelaziz and Mr. Wilson found the weak spots in the government’s case, parents who pleaded guilty are unlikely to be able to challenge their convictions on similar grounds, legal experts said.

Mr. Abdelaziz was accused of paying $300,000 in 2018 to have his daughter admitted to the University of Southern California as a top-ranked basketball recruit even though she did not make the varsity team in high school.

Mr. Wilson was accused of paying $220,000 in 2014 to have his son admitted as a water polo recruit at U.S.C., even though prosecutors said he was not good enough to compete at the university.

Mr. Wilson was also accused of agreeing to pay $1.5 million in 2018 to have his twin daughters, who were good students, admitted to Harvard and Stanford as recruited athletes.

They were tried together in the fall of 2021; Mr. Wilson was later sentenced to 15 months in prison, and Mr. Abdelaziz to a year and a day. Their lawyers argued that the men thought they were making legitimate donations to the university. They said they trusted Mr. Singer, as their college consultant, to guide them.

The investigation ensnared more than 50 people, including the actresses Felicity Huffman and Lori Loughlin; Ms. Loughlin’s husband, Mossimo Giannulli, a fashion designer; and coaches and exam administrators, among others.

Mr. Singer agreed to cooperate with the government and pleaded guilty in 2019 to conspiracy charges. He was sentenced in January to three and a half years in prison.

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