As the city Department of Education reevaluates school district zoning in Lower Manhattan, parents in the region have handed their support to a proposal that would guarantee kids a spot in the schools closest to their homes. John White, head of the department’s office of portfolio development, said that parents are eager to form school districts based around neighborhoods. The other proposal would create a larger school district, in which students would have their choice of different schools from different areas in the region. The District 2 Community Education Council is rezoning the neighborhood as two new schools open to relieve the persistent crowding in Tribeca and Battery Park City schools.