Does AA have access to all the Qantas award space listed on Qantas.com?

I'm hoping to fly in Qantas First on the A380 direct from Melbourne to Los Angeles. When I called AAdvantage to book, I was looking at qantas.com. It showed availability on Nov 29 and Dec 4. The AAdvantage agent said there was no space in first those days. She did find the October 31 space I saw.

What gives? Does or does not AA have access to all space shown on Qantas.com? If not, how would I know without calling AA which space AA does have access to?