State tax filing

No.

If you maintained a permanent home in NYC and were simply visiting your parents, you would not have a PA tax liability unless you also earned income from Pennsylvania sources. You say you "never established residency in PA" so you would need to show where you did have residency during that time. 

The fact that your tax documents were sent to a PA address has no bearing on your tax liability. It does not matter that your parents address is on your W-2.

HOWEVER the fact that the W-2 shows PA income in box 16 could be a problem. 

Pennsylvania will get a copy of that W-2 and be looking for a state tax return from you. 

The best remedy is to have the employer file a corrected W-2 (W-2c) showing only NY income. 

If that is possible, have the employer issue a W-2c, that would remedy the situation.

If that is not possible, and if PA tax was incorrectly withheld, you could file a PA state return to get the tax refunded, but you indicate that no PA tax was withheld, therefore the program won't let you file a zero PA state return with no tax liability and no refund due. 

• Therefore, if a W-2c is not possible, you can do nothing and when you get a letter from the state of Pennsylvania send proof that you resided in NY.
• You could also mail in a paper PA state return with a copy of the W-2 and an explanation that the income was erroneously reported as PA income and that you only had your mail sent to your parents home as a convenience.  
• Lastly, if you did not have a place in NY while you visited your parents, you WOULD be a part-year resident for both PA and NY in which case you would need to file that way. Do the state you were in on Dec 31 first, then the other.

Make sure all the income is accounted for since it sounds like the severance pay had no withholding for either state. 

I don't understand the entry for local wages, is it a negative number? Which locality? 

If you continue the thread, please clarify if you maintained residency in NY the entire year.