The new plan, promoted by Mayor Bill de Blasio, was supposed to take effect for the 2019-2020 school year. The suit asks the court to put it on hold while the case is considered.

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The dispute has largely been a political question, with New Yorkers forced to consider whether their system gives an unfair advantage to one minority group over another, and whether better integrating the city’s best schools is more important than maintaining uniform admission standards, determined by one high-stakes test.

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Now, the question moves to the courts, which have set a high bar for schools seeking to achieve greater racial diversity.

The city’s plan would significantly expand its Discovery Program, which admits some students who did not get into elite schools through testing. But it is limited to the most socioeconomically distressed middle schools. The suit alleges that schools with large Asian American populations were purposefully excluded from this formula.

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Today, the Discovery Plan admits only about 6 percent of students attending the elite high schools. If fully implemented, 20 percent of seats at each school would be set aside for students from designated low-income schools by fall 2020.

The suit, filed in federal district court in New York, argues that the Discovery Program was designed to help low-income students from across the city and that the city’s move to restrict the program to certain middle schools represents an unlawful effort to manipulate the racial balance.

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Will Mantell, a spokesman for the school system, said the changes would nearly double representation of black and Hispanic students at the elite schools, to 16 percent.

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Today, black and Hispanic students make up 68 percent of the city’s high school population overall but just 9 percent of students offered seats at specialized high schools. At the same time, 62 percent of seats in elite high schools are occupied by Asian American students.

“Our reforms will expand opportunity and raise the bar at our specialized high schools,” Mantell said. “Our schools are academically stronger when they reflect the diversity of our city.”

The plan, announced in June, triggered waves of angry protests from the Asian American community, with parents arguing their numbers should not be limited if their children are the most qualified.

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The lawsuit’s plaintiffs include Yi Fang Chen of Brooklyn, the mother of two young boys. She alleges that the new system will put her sons at a disadvantage if they choose to apply to these schools one day.

“I truly believe in merit. I think there should be equal opportunity for all,” she said in an interview. “Diversity is a great thing, but do not lower the standards.”

Chen immigrated to the United States with her parents as a teenager and earned a doctorate at Stanford University.

Chen said she joined the suit after talking with other parents about their frustrations. “We tried protesting, and that was not helpful,” she said.

Other plaintiffs include the Parent Teacher Organization at I.S. 187, also known as Christa McAuliffe School, a predominantly Asian American school that sends a high proportion of its students to specialized high schools. Of 274 eighth-graders who graduated from the school in 2018, 205 attend specialized high schools, more than from any other middle school in the city, the suit says.

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Most students at the school come from low-income families, the suit says, but the school was not considered poor enough to qualify for the expanded Discovery Program.