According to the NY dept. of revenue web site, NY does give a credit for local, as well as state, income tax paid in another jurisdiction.  I'm not specifically familiar with the NY software, but that typically comes up on the credits screen and is known as the resident credit.

Resident Credit

Who is eligible? 

You are entitled to claim this nonrefundable credit for the tax year if:

• you were a full-year or part-year resident of New York State,  and
• you had income sourced to and taxed by another state, a local government within another state, the District of Columbia, or a Canadian province 

https://www.tax.ny.gov/pit/credits/resident_credit.htm