As New York City reckons with the fact that its schools are among the most segregated in the nation, a swath of Brooklyn synonymous with fiercely progressive politics is about to test how effective a local integration effort can be.

The plan, in District 15, which includes Park Slope, Sunset Park and Red Hook, has been pushed forward by parents, rather than by the de Blasio administration, which so far has been unwilling to propose a systemwide desegregation strategy. And it comes in one of the rare districts that has a diverse enough population to make integration feasible.

Richard A. Carranza, New York City’s schools chancellor, is poised to approve much of the plan before school starts next month. If he does, it would allow the district to replace a competitive admissions process in which middle schools screen fourth graders based on test scores, attendance records and even the occasional recommendation letter, with a lottery. The plan, which could go into effect as soon as this fall, would also require each of the district’s middle schools to reserve about half of its seats for students who are from low-income households, homeless or learning English.

In an interview, Mr. Carranza said that District 15’s plan highlights the potential of parent-driven efforts to inch the city forward. In the last two years, parent councils on the Lower East Side and the Upper West Side have had their own desegregation efforts approved by the Department of Education, sometimes after extended periods of private negotiations and public outcry.