There are some schools in all boroughs that are "unzoned". Some accept children based on academic achievement, some based on talents (usually in the upper grades) and some by lottery. Take a look at
www.insideschools.org find a school feature to find those schools in your borough and/or neighborhood. You can also search for charter schools in that way. In the case of the unzoned schools, the Region makes the decision re placement. The charter schools hold lotteries, usually in the spring. You have to contact them directly.
If the schools the children are in now are on the No Child Left Behind list, (NCLB) there is an option to transfer to a better school
in the Fall. The choice is not open ended, however, you will be given a list of available places and you must apply for the best one for your children.
Outside possiblity: make a case for the health of the children and/or the child care needs. But be aware, there has been little movement on that score in recent years.