Flagship lounges are usually much smaller than Admirals Clubs. They offer free liquor and usually have a better selection of snacks than ACs do. As mentioned FLs are reserved for 1st Class int'l passengers and OneWorld top-tier int'l passengers (including AA EXP). I find that due to their size they are often more crowded than the AC, so unless you want to drink they don't offer much more.

BTW, if you are flying Biz, you are entitled to enter the AC. I would take a guess that if your parents enter the FL, the agent there might let you join them.