You should think about your aging parents back home.
Literal
Hometown [possessive-の] aged parents [possessive-の] things [object-を] think [try-てみる] should.
老いた ('aged') is the resultative form of 老いる functioning as an adjective. 故郷 (ふるさと/こきょう, 'hometown'). 考えてみる ('to try thinking') uses ~てみる for tentative action. ~べきだ adds obligation.