It is illegal for persons under the age of 18 to get tattooed regardless of the presence of parental permission in the state of New York. Some states allow minors to receive tattoos if they have the consent of a guardian, but New York law completely prohibits this.
With parental permission, minors are allowed to receive body piercings in any place in New York. Piercing of the ear does not require permission, but any other piercing requires written consent. The New York Department of Health has a full list of laws and requirements for piercing, tattoos and other forms of body modification.