NY can be a very scary place. Once I crossed the street at a zebra crossing (so I had the legal right of way), but this taxi driver glared at me (for honsetly abosultely no other reason than he had to stop to let me cross - as did lots of other cars - but I did NOTHING to justify his actions, I didn't even look at him for more than a second) and when I got onto the sidewalk he drove up onto it and chased me down the sidewalk (I truly had to run as there were no doorways to duck into)
So anyway, take your parents advice and heed their warnings, they are older and wiser than you think
Good luck

... and have fun, I do love NY!