Sorry, but there are some factual issues here which do not help your parents. I'll be blunt, because sugar-coating won't help them.

They were presumably traveling in Y (they purchased extra leg room). The LX check-in deadline at JFK is T-75 for Y passengers. Thus, even had there been no lone, they would likely have missed check-in and even a very short line meant they were definitely too late. LX recommends arriving 3 hours prior to departure. That is not a requirement, but it is hard to argue about lines and the like when you do not follow the recommendation.

At T-75, LX would have cancelled your parents' reservations. That happens on pretty much all carriers and is likely automated. What happens next is dependent on your parents' fare rules. If these were standard discounted penalty fares and all of this was caught before departure, they are subject to paying a penalty + fare difference to be rebooked. While it would have been nice of LX to have waived the fees or the fare difference or even both, it is entirely discretionary and not typical of LX to have done so.

The $1,600 is likely made up of roughly a $300 fee and a $500 fare difference x 2. You can look at the fare rules and assure yourself that the $300 is correct and also look at their original and new fare bases to figure out whether the fare difference makes up the rest. The numbers your parents were charged are certainly in the ballpark.

While I doubt that this will do any good, they should write (perhaps you can help) a clear & concise letter explaining what happened. But, they need to get the facts much more clear than in your OP. Arriving at the last second as you say they did would have been a problem for an Olympic athlete as much as for an elderly traveler. If the numbers are correct, theirs is a plea for mercy. If the numbers are wrong, then it is a demand for a refund.

Finally, I see the entire Expedia and who is responsible for assisting as a red herring. It doesn't matter and the fact is that when you called, LX helped out.

Not to berate you or your parents, but because this is a public forum, cutting deadlines too close is always problematic. It is even worse when someone is inexperienced and perhaps a little less able to sprint around. Better to take the carrier's suggestions than to show up as check in is closing and hope that there will be no line of any kind.