doctor who - Is Peter Tyler's Death a Fixed Point in the Timeline?

In Father's Day Rose asks the Doctor to take her to when her Father died, so he won't have to die alone. After her Father is hit by a car, Rose is overwhelmed and can't go comfort her Father. Rose pleads and the Doctor takes her back, and they see the first set of the two of them watching her Father. The Doctor tells her, "It's a very bad idea having the two sets of us being here at the same time."

When the car is heading toward her Father, Rose, on impulse, runs out into the street, in front of the earlier versions of her and the Doctor and saves her Father. As she does this, the earlier Rose and Doctor vanish. They don't vanish when they see her, but when she prevents her Father's death.

Then, later when they're all trapped in the church, the car that hit her Father keeps appearing out of nowhere, driving along the road, and disappearing. This is the road near where they're hiding, not the one where he should have died. The car continually appearing and disappearing is like a break in causality, as if it's the start of everything happening at once, like we see in The Wedding of River Song.

Time is only set right with the death of Rose's Father, not with the resolution of the paradox of the earlier Rose and the Doctor disappearing or Rose and the Doctor having been in the same place twice. Much of the situation seems to revolve around Rose's father being alive and not dead, rather than she and the Doctor being in the same place at the same time twice.

Is her Father's death a fixed point in time that, when changed, upsets time?

Answer

The "dead-man-walking" was the paradox, not the (Doctor+Rose)x2.

It seems Peter Tyler's death is an integral part of the natural timeline. Most likely because had he lived, Rose might never have met the Doctor, and Bad Wolf would not have played out the way it did. Had that transpired, the galaxy might have been a very different place. Also, parallel world Peter Tyler played a part in the invasion of the Cybermen, which too, would have changed the timeline drastically had he lived.

So if what you're asking is whether his death is an inelastic point in the timeline, then yes.

Time in the Doctor Who universe appears to be like a rubber band that is anchored to fixed points at certain critical places. You can pull and stretch it, but it'll snap back. Pull too hard and it breaks catastrophically.