N.Y. Educ. Law § 4812
The board of managers may make an agreement with the parents or guardian of a child in such school for the payment of an amount therein specified for the instruction and maintenance of such pupil. An application for the admission of a child with the consent of the parents or guardian shall not be granted unless suitable provision be made for the clothing of such child. The amount agreed to be paid for instruction, maintenance and clothing shall be secured to the satisfaction of the board of managers. Such board shall ascertain by investigation the financial ability of parents, guardians and other persons legally liable for the support of pupils admitted to such school upon commitment, and may demand of such parents, guardians or persons the payment of an amount reasonably sufficient to pay all or a portion of the cost of instruction, maintenance, and clothing of such pupils. The board may proceed against such parents, guardians or persons, by proper suit or proceeding in a court of competent jurisdiction for the recovery of the amount agreed or required to be paid, as herein provided. The amount so recovered, after the payment of the proper costs and expenses of such suit or proceeding, shall be paid into the treasury of the city, and shall be applied to the payment of the cost of the instruction, maintenance, and clothing of such pupils. Such action at law shall be brought in the name of the city by the corporation counsel thereof.
This provision shall in no way affect the power of the committing court to order payment as provided in subdivision b of section nine hundred twenty-one of the code of criminal procedure.
Children committed either by court or school authorities of the city shall be a charge upon such city. Children so committed from outside of the city shall be a charge upon the city, incorporated village or township from which committed.
N.Y. Educ. Law § 4812