Alright hear me out on this. This is about Zero Squad. Back in the first flashback episode they just mentioned Zero Squad. The old 12th squad commander got promoted to become the new commander there. Well they never said her name or have shown any pictures of her.....Here it is. I think it is Ichigo's mom.
When a spirit leaves a body, people think that they are dead. A perfect example of this is when they first introduce Don Kanoji. They find Ichigo's body and think the worst.
Now as far as Ichigo's father I think he is the VICE-captain of Zero squad. This explains the paritail white coat. Not quite a full one, but could also be symbolic of a vice captain of zero squad.
What do you think????