You don't have to tell your parents
But you'll have to come up with a reason why you'll be gone for so long. Court can take all day.
In New York, if you're 16 or above, you're charged as an adult-so no parents need to be with you in court.
It sounds like you were charged with Unlawful Possession of Marijuana (PL 221.05), which is a violation and not a crime. The maximum fine is $100. It also sounds like you've been told to appear at 346 Broadway, which is the "summons part." Am I correct?
If you have time, I would attend proceedings at the court where they summoned you to appear, so you could get an idea of what they offer first-time offenders. I would also inquire as to the availability of lawyers, and whether a lawyer will automatically be there to advocate for you (I'm not really sure what goes on in the summons part).
I do know that If an "ACD," is offered, I would take it. Just stay out of trouble for a year, and the case will be dismissed and sealed. It will not become a part of your permanent record if you stay out of trouble for that year.
Good luck