Suzanne is right in that often times airlines change partnership and if you buy long in advance, you might not get credit for miles flown if particular airlines are no longer partners. Good luck in getting the miles credited. There are all sorts of strange partnerships out there. It does not matter which airline issued the tickets since most of the airlines have "interline" agreements where they can all issue each other's tickets much like a travel agent. There are strange mileage partnerships out there though - ie you can earn Delta miles (not elite qualifying miles though) by flying Singapore Airlines even though Singapore is part of Star Alliance while Delta is part of Skyteam.

Sheri, US cannot "see" if you've flown on different airlines since its not their system. UA was ridiculous in telling your parents to go talk to US to make sure their miles get credited to US since there is nothing that the US ticket agent can do to ensure that, the UA agent just needed to make sure that their US numbers were in the Frequent Flyer number field.

Boarding passes and ticket receipts are important things to hang on to since without them the airline that you want the miles on has no way to be certain that you actually flew and at a mileage earning fare (all US based airlines give miles for all revenue tickets but many of the international ones particularly the Asian ones do not give miles for "deep discount" tickets).