Michael Mulgrew, the union’s president, said the city needed more school seats and called for better planning by officials to address fast-growing neighborhoods such as Lower Manhattan, Downtown Brooklyn and Hunters Point, Queens. “New towers are all over the skyline, not just in Manhattan,” Mr. Mulgrew said. “And new buildings mean more families and more pressure on schools that are already at or over capacity.”

City education officials said they planned to add more than 44,000 new school seats across the city in coming years. In Lower Manhattan, eight new schools and an annex to an existing school have opened since 2007, adding more than 5,600 seats and helping to maintain, or reduce, the average class sizes. “We are dedicated to addressing overcrowding,” said Devora Kaye, a spokeswoman for the Education Department, adding that it was working closely with families and the community to identify new school sites.

Several of these new schools in Lower Manhattan are already grappling with space issues. The Peck Slip School (P.S. 343) in the seaport, which opened in 2012, has grown to about 380 students — too many to fit on its rooftop play area at one time. So the school has lobbied to close the street it is on during school hours to use as a playground. LAZ Parking, which has an entrance on that block, initially opposed the request, but later agreed to the street closing after Manhattan Community Board 1 and elected officials negotiated a compromise in which a new entrance to the parking lot will be created on another street, said Anthony Notaro, the board’s chairman.

At the Spruce Street School (P.S. 397), which is in the new Gehry building, parents have raised concerns about a parking garage overseen by NewYork-Presbyterian Hospital that opened this year next to the school’s entrance. Now cars entering the garage drive through a plaza where parents and children walk. Though barriers have been erected to separate them, many parents find these inadequate. “It’s a big concern,” said Mr. Proulx, of the neighborhood association, whose three children attend the school. “You have an unsafe condition because of the conflict that is inherent between children and automobiles.”

NewYork-Presbyterian said it had worked closely with school administrators and community leaders to maintain a safe environment around the garage, including the installation of a traffic light at its entrance, allowing only trained attendants to drive the vehicles and activating an alarm system during school fire drills when children can walk in front of the garage.

“We remain committed to having further discussions with the school, parents and all of our neighbors in the interest of protecting everyone’s safety,” said Karen Sodomick, a spokeswoman for the hospital.