Had a similar-ish problem earlier in the year. Hard one as there's no magic solution.

Are cash tickets available on the LHR-ATH leg? If so, assuming your cancelled seats don't go back into award inventory, surely that is an option? Although Club Europe will probably be expensive, personally I don't think it's such a big deal to take a reasonably short flight down the back, given they'll have access to the highest levels of lounges thanks to flying inbound in F (assuming it's same-day), and will have just done LAX-LHR in the upmost luxury

If there aren't even any cash tickets on BA available for LHR-ATH, then you're in a tight spot, as you'd need to start looking at alternative LHR-ATH carriers such as Aegean or Olympic which would involve a terminal change. At this point you have to ask what your parents would prefer: a) Club for the long flight with an easy connection plus Club for the short flight, or b) First for the long flight, with a trickier connection, and probably no lounge access for the onward (unless you splurged on business tickets).

My suggestion would be to see if cash tickets are available on BA for the LHR-ATH leg. If they are, and they are affordable, there's your backup in case the seats don't go back into award inventory (and they probably won't).

If BA are completely sold out, I'd probably avoid the hassle, cancel the F tickets and send them in Club as per your original plan.