Originally Posted by SEA1K4EVR

They should bring a print out of their United itinerary with them to the checkin counter for their Cathay flight. The Cathay agent *may* tag their bag all the way through to their final destination. If they can do it then you're right your parents will just be able to drop the bag off at the United bag drop area and be on their way. But Cathay and UA are not partners and it's possible the Cathay agent will just tell your parents to claim their luggage and re-check with United in which case there would be a bag check fee. Probably one of those things they won't be able to find out for sure until the day of the flight.

As far as the connection process.. after dropping their bags off they'll either need to walk or take the Airtrain to terminal 3 and re-clear security there for the onward United domestic flight.