Call BA until you get an agent who will play along.

What you want to get is a oneworld award. It's based on total mileage (by segment, not from point-to-point, I think) and your parents will need to fly 2 other carriers other than BA (most likely AY, CX & QF).

Suggested itinerary:

BA U.S.-LHR
BA LHR-Edinburgh
BA Edinburgh-(LHR)-AMS
BA AMS-(LHR)-MXP
or IB AMS-(MAD)-MXP
AY MXP-HEL
AY HEL-SVO
BA SVO-(LHR)-NRT
QF NRT-(SYD)-AKL
QF AKL-SYD-LAX

parentheses means transits and no stops.

FWIW, I flew YVR-HKG-BKK-MEL-LST-MEL-SYD-ARM-SYD-HKG-NRT-YVR last year on such an award issued by CP.