Just chiming in to say that as far as I know airlines (at least in Canada), for security reasons, cannot knowingly let a flight go carrying checked bags for passengers who are not onboard. I'm an FA and we often have to delay flights when passengers who are checked in "go missing" in the airport so that the ground staff can find and remove their bags from the cargo hold. I agree with the other poster that if this is something the airline finds out was intentional, it could earn your parents a black mark next to their name with that airline. I believe there are exceptions to this and sometimes bags are sent on flights in error but if they were to travel with their bags checked under their name, the second leg of that journey *could* be negatively impacted if they just choose to go home at that point. If they only had carry-on luggage, that should solve that issue.
The rules for airlines in other countries may vary (I'm too lazy to look up the ICAO regulations) so don't take what I'm saying as the absolute rule. I hope you find a solution!