Ms. LaRosa, 44, chose S.T.A.R. “In S.T.A.R. Academy, I feel like my voice is heard,” said Ms. LaRosa, who is Hispanic. When she interacts with Neighborhood School parents, she said, “I feel like I have to talk a certain way or express myself a certain way, because I feel like I’m being judged.”

In contrast, Ella and Brett Leitner, who are both white, gravitated toward the Neighborhood School, where students call teachers by their first names and have a say in deciding whether the class will study, say, the civil rights movement or the debate over the Dakota Access pipeline.

“For us the social-emotional piece was such an important metric of what made a school good,” said Ms. Leitner, 43. If they had not been able to choose a progressive school, she said, they might have tried to get a scholarship at a private school or sent their children to a traditional school and pushed for a more progressive curriculum.

The district started the progressive schools in the late 1980s and early ’90s, in an effort to stanch a steady loss of students from the district, mostly to District 2. (It also allowed students from outside the district to enroll, and today 30 percent of elementary school students live elsewhere.) Hispanic, black and Asian families apply to the progressive schools at lower rates than whites, and the schools are all whiter than the district as a whole, with fewer low-income students.

Bradley Goodman, the principal of the East Village Community School, said the imbalance was “something that we’ve been thinking about and talking about for a long time and wanting to address.” His school is 55 percent white. He and the other principals from progressive schools have been part of a diversity initiative that let them set aside a certain number of seats for students who qualify for free or reduced-price lunch or who are still learning English. The four schools met their targets this year, though both Mr. Goodman’s school and the Neighborhood School made offers to several families who ranked their school as a fourth, fifth or even sixth choice, while turning away other families who ranked it first. But drawing more black and Hispanic parents may be tough.