I am more of a Delta guy, but came to AA and BA miles with credit cards.

I am looking to get my parents from YYZ-HKG in the very near future, and while I understand the general availability of flights in this market, I am confused about finding specific flights - YYZ-HKG direct on CX. It seems that while availability is not high on BA, there are some availability. For example, I can find YYZ-HKG direct on CX on ba.com April 3 and a couple dates in Feb. However, I was not able to get anything to show up on aa.com from feb to may.

Am I doing something wrong? Or is availability of CX award space that bad on AA? Since the flight originate in Toronto (North America), I shouldn't have to call?