If your father, who is already married to your mother, does not open the car door for her, it has nothing to do with tznius. He just doesn’t open the car door for her. There are ways to open a car door even from the inside by leaning over the seat and pushing it open. He could have figured out like others have said to open it and keep walking, etc. So your father just doesn’t bother or never gave it a thought and it has nothing to do with hashkafa.
Be honest, did you actually have this discussion with your father?