Most U.S. school districts draw "attendance boundaries" to define catchment areas that assign students to schools near their homes, often recapitulating neighborhood demographic segregation in schools. Focusing on elementary schools, we ask: How much might we reduce school segregation by redrawing attendance boundaries? Combining parent preference data with methods from combinatorial optimization, we simulate alternative boundaries for 98 U.S. school districts serving over 3 million elementary-age students, minimizing White/non-White segregation while mitigating changes to travel times and school sizes. Across districts, we observe a median 14% relative decrease in segregation, which we estimate would require approximately 20% of students to switch schools and, surprisingly, a slight reduction in travel times. We release a public dashboard depicting these alternative boundaries and invite both school boards and their constituents to evaluate their viability. Our results show the possibility of greater integration without significant disruptions for families.