I am P.O.A. of my mom's money. She has Parkinson's. My brother is the executor of my dad's will (Living Trust) which assigns my brother as Executor. My dad's money will be left to my mom. If my dad passes away before my mom, will I remain P.O.A. over the finances that my dad leaves to her?

The simple answer is yes SS.
But his work isnt finished so quickly, dependent on the particulars of the Will and living trust, also the closing of that estate (your Dads) there could be a period of time when you both retain some kind of legal obligation together.
Your POA doesnt end till TOD then all powers are immediately void unless mom chooses to replace you with someone else.