Two comments.
First, I recall a study some years ago relating parenting styles to metaphorical role descriptions, in this case “strict father” versus “nurturing parent”. I mentioned this to my wife, a mental health counselor, and she said her field categorizes three types, adding one for total license (per Martin’s example). In the study I mentioned, the children brought up under the “strict father” style did not tend to turn out well, tending to become anxious and indecisive as adults. The children brought up under the “nurturing parent” style tended to be confident and unafraid of making decisions.
Second, it may be useful to analyze the ethics of these two styles on a “means” versus “ends” ethical basis. From my own episodes of weakness and indecision as a parent, and from observing other parents in the situation of publicly screaming at their children, it seems to me that the parents in those situations are mainly concerned that their children, and their lack of control over them, reflect poorly on themselves. That is, they see their children somewhat as means to their own social standing. A fully nurturant parent (assuming such an entity can exist), on the other hand, is fundamentally concerned with the child’s well-being.
In the situations of which I speak, concern for the child’s well-being includes the child’s development as a socially competent individual. An ideal nurturant parent would attempt to use such situations to further such aims to the extent possible. Doing this effectively means tuning-out the real or imagined disapproval of bystanders and acting with composure and self-confidence. The study I originally referred to suggests that this will be more difficult for parents brought up under “strict father” or “total license” models than for those brought up under “nurturant parent” models.