You don't need a copy of the reservation. They have a list of guests checking in each day.
OP, don't over think this. When you pull up at the Grand, the gate guard will tell you exactly what to do. They will literally walk you through every step. They will ask you if you are checking in. You say yes and give them your name (they might ask for an ID card). They will ask you if you want valet or self park. If you say self park, they will hand you a slip of paper to put in your dashboard. They will tell you where the lot is and how to enter it. They will show you where to pull up to unload your bags. You will unload and then either hand bags off to bell services or leave them with a traveling companion while you park the car. You walk back across the street, and show your photo ID to the security guard, letting them know you just arrived to check in. Then you check in at the desk. Your parking charge will be reversed since you are
DVC on points. When it's time to leave, all you do is show your room key to exit the lot. You can stay parked there until midnight on your check out day.
FYI, when we arrived 2 weeks ago, they were not allowing people to park in the port cochere while checking in. It was load/unload only due to the construction work they are doing. Plan to park BEFORE checking in if you are arriving in the near future.