According to the
AA website, AA Platinum members are entitled to:
# Entrance to all of the oneworld airlines' Business Class airport lounges, regardless of the class of service flown when traveling internationally3
3AAdvantage Executive Platinum and Platinum members must be departing on a oneworld operated/marketed flight that day. [...] Lounge access may not be available due to capacity constraints during peak travel periods or at some lounges operated by 3rd parties.
If your flights are on LAN airlines or LAN Peru (but not LAN Ecuador or LAN Argentina), you will have lounge access everywhere there is a OneWorld lounge.
Enjoy your trip,
Michael