When barbara asks her father why she is not allowed to play in the street, he says, "because i say so, and if you don't do what i say, you will get a spanking." this illustrates the style of parenting?

It's a dominating parenting style. It doesn't give the child an idea of what dangers there are, and it makes the child more afraid of punishment than understanding that her father is merely protecting her from being run over.

Her father is basically saying that if she disobeyed him it'd hurt his ego as a father who should be obeyed  more than actually caring for her safety and well being.  This is what the proper scenario should be " Daddy, why can't I go play in the street?"

" Because honey, there are cars that go down that street, and a little girl like you wouldn't be seen that easily and you could get hurt. I tell you not to go play in the street because I love you and I want you to play somewhere safe.  Now that you know what would happen to you when you play in the street, and that I told you not to, do you know what'd happen to you if you did it anyways?"

And explain the punishment and why she'd be punished. When setting rules and when carrying out punishments it should NEVER be in anger. It shouldn't be as if they wounded your pride as the one in charge over them, it should be because they scared you when they did something dangerous, even after telling them to never do it.  Set age appropriate punishments

For the first offence, a small punishment of having a time out.

Second offence, punishment as in x amount of time of no privilege with an item or event.

Third offence, A spanking. This is last resort. Do it in any manner you wish so long as it's appropriate and not prolonged to leave a mark.

Spankings are punishments if the child has been explained a real reason of why they shouldn't do something, yet ignore the possibility of being seriously injured and do it anyways, resulting in a mild punishment.

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your welcome.