Normally in Massachusetts if a parking ticket goes unpaid past 21 days an additional charge (usually $5.00) is applied and a notice is sent to the registered owner, if not paid after the first notice the jurisdiction where you received the ticket will then send out a second notice.
If the parking ticket is not paid the violation is then forwarded to the Massachusetts Registry of Motor Vehicle for non-renewal of the registered owners drivers license or motor vehicle registration, however since you are not a MA licensed driver these penalties would not apply. Your driving privileges in Massachusetts though may be suspended since you are an out of state licensed driver and they therefore cannot restrict your registration.
In Boston the Office of the Parking Clerk (OPC) is responsible for the collection and adjudication of fines and fees resulting from the issuance of parking tickets, so you should contact this office to see what penalties may arise for a New York licensed driver if they either paid a parking ticket late or failed to pay it at all.
New York does not normally place out of state minor tickets, including parking tickets, on your NYS driving record or assess points so if you did pay the ticket it should not affect your New York driver's history. If you do not pay it though it could continue to affect you in MA and the payment amount can continue to grow.