The good news is that Thai Airways is part of Star Alliance, however, you didn't leave much time for their connection. They can show their connecting itinerary at their originating airport, however, the computers may not allow the connection to be printed on the bagtags.
Unfortunately, the issue with timing will be how full the flight is and whether it arrives with other big jets. If your parents are flying upfront with bags tagged for priority, it could still take an hour just to clear immigration/customs. The problems with delays tends to be how full the halls become when multiple 747s arrive at the same time.
Whether or not UA flight is delayed, won't help if your parents have to check bags. LAX is known to cut off baggage checkin up to 45 min prior to departure.
Your best bet is to have them ship their luggage and carry on essentials. If you have their itinerary, you should be able to check them in online for their connection to secure their seats. they can always have their boarding passes reprinted at the ticket counter.
BTW, welcome to FT.