I see nothing for a table of six for that day online for any meal, which probably means that they don't have any room for party of six in the restaurant. You could try to show up with an extra two, but I wouldn't get my hopes up on being seated promptly, if at all, at least together with the entire group.
I suggest that you tell your parents to make precautionary reservations somewhere nearby, like the Kona or Grand Floridian Cafe for a half hour later than your current reservation, so they have somewhere to go if they get turned away at the door, which I'm sorry to say is pretty likely.