Why do I owe NY tax? I was a full time student and graduated in June of 2017

Did you have New York income or were you a resident of New York?

I am a resident of NY. (Not city). I worked oartine during year with a income of $5,900

Partime

Were you claimed as a dependent on your parent's return?

Yes

If you are single and claimed as a dependent, you will likely owe New York tax if your income is more than $3,100.  Because you were a dependent, most of the state credits do not apply.  New York has the reputation of one of the highest taxing states for a reason.