SK requires passengers to be at the gate for boarding no later than T-20 (T-25 for US flights). Presuming that this was xFRA, it will matter whether the family arrived at T-21 or T-19. If at CPH, they were 5 minutes late.

Most carriers have deadlines and the SK deadline is actually a bit better than many carriers. The deadline is designed so that the aircraft can actually push on schedule. It takes a goodly amount of time for passengers to get their luggage stowed, themselves seated, the crew to do safety checks, the Captain, gate and dispatch to make the numbers for weight & balance as well as fuel work.

If OP's family were on discounted tickets which were non-refundable, they likely were done a solid by the agent who helped them. Often, the result is that the tickets are cancelled and all value, sometimes save taxes, is lost. While the loss here is clearly painful, it could have been worse.

Thus, this comes down to whether the family made it to the gate in time. If not, why were they late? Is there any reason which might permit a claim on their travel insurance? If not, this is hard and all they can do is, after their return, perhaps send a short note to SK admitting fault and asking for mercy. The chances of mercy are slim, but worth the cost of an email.