" ... in the system."
A 1958 vehicle ... maybe ... last registered in the 1960's is still (good news) in the system.
Somewhere.
Probably in a cardboard file box in a dusty warehouse in Albany (bad news).
The car may have lived its entire life in the age of paper documents. If it made it to NY's early computer base, unless it remained registered with a tag, it eventually was purged from any conversion update to the digital system.
Either contact whatever tag agency (DMV) and simply tell them you're trying to locate your folk's old car. Sometimes being truthful works better than BS. Or, most police officers can trace a current vin on a laptop in their cruiser.
This all opens a related subject. If you have an untagged, old title, whether or not you have the matching vehicle, that title could be purged from the system and leave you with a lot of red tape to contend with if you wanted to register the car/title.