First of all, I'm a bit confused as yours is for half an hour starting at noon and your parents for an hour starting at noon -- I thought they were only for half an hour.
But, like any fastpass, that's just the time in which you need to show up. So you could all show up at the same time, assuming your passes overlap, and you'd be fine. There's no assigned seating -- I described it to my friends as fast food in a steakhouse environment. Yes, the silverware is 'real' - but the napkins are paper and you have to go pick both of them up. Yes, the food is delivered to you - except for your drink, which you fill up in a paper cup at a drink station exactly like the one McDonald's or Hardees or wherever would have.
So, as long as your passes either overlap in time or theirs starts right when yours ends, you should be fine. (In that latter case, you'd need to check in towards the end of your time, waste a few minutes looking around - maybe you can use the restroom, I'm not sure - and then they show up at the beginning of their time and you all pay at the cash register one right after the other.) It's not until you pay at the cash register that they put the food together (sheesh, I hope not!

) so if you can finagle it such that you are all together in a group by that time, you should be fine.
You check in at the 'bridge' entrance; when I did it, the fastpass people were in one line and, I guess, the folks that showed up early enough were in another line. Alternatively, it could have been that I was put in one line because I'd already ordered while the other line was for those that had not made their decision. I'm not quite sure which, to tell you the truth. Regardless, you walk across the bridge to get inside and then you pay, then you get your drink(s) and silverware and find a place to sit. When the food's ready, it uses a plastic 'rose' they've given you (if you don't have the magicband) or your magicband to find where you've seated yourself. So even if they're on a different reservation, if you're seated together, the food will come to you just fine.
It's an interesting experience. I hope you enjoy!