Infrequent OW flyers - which programme?
 

I booked my parents on one of the cheap CMB-JFK deals in business class.

As part of this they are flying CX from AKL-HKG-AMS. Normally they fly MH or Star Alliance, so they won't be racking up many more miles.

Which programme should I sign them up for? Asiamiles? AA?

There isn't much point in optimizing orphaned miles; presuming they won't have enough to redeem for an award and little prospect of topping it up later. So consider a plan which lets them combine miles (e.g. QFF family plan, which would let them double the miles in one account). Asiamiles has the most upgrade capability and partners with some interesting non-OW airlines, so maybe that would be the best place for parking a small number of miles if not shooting for a specific award. I think the award costs just went up by a lot (at least for some awards) making it less attractive. When in doubt AA is usually the best.