It's been a good while since I've done any chocobo breeding, but if I remember correctly, it uses the ratings of the grandparents to determine the rating of your chick. So if you want four stars in a certain stat, all of the grandparents have to have four stars in that stat to guarantee it being passed on. This is why you usually end up breeding multiple birds at a given pedigree level and crossing them with each other to lock in star ratings. It's a slow process unless RNGesus really likes you.

The hereditary ability the chick gets can come from either parent, so it's 50/50 you get the one you want.

They're also no reason to buy a P9 permission if you're breeding a P2 bird. Higher pedigree permissions do not guarantee better stats. It's all luck of the draw.