Your parents should be able to accumulate miles in the American AAdvantage program in their own accounts (they must open accounts and have their frequent flyer numbers added to their itineraries. AA does not allow family accounts.
You should visit the JAL forum to see if JAL allows family accounts, etc. If they reside in Japan, they might benefit more that way.
Both AA and JL are airline partners and belong to the same alliance (oneworld).
As US and AA have not integrated their programs, we'll eventually move this to the AA Pre-Merger Forum. Moderator