CNY, analogies that make it hard to argue your point aren't useless - the relevance here is that, as you describe in your post, it would be no defence to a crime to say "eh, I didn't hit my wife/stranger/sister that hard, I used a looped belt."
Yet that becomes a defence for belting your 16 year old disabled daughter. Now, it's illegal for adults to do it to each other for good reasons. Why aren't those reasons just as good when applied to a 16 year old?
Rev made the point that there's a relationship of trust and guidance, giving the parent responsibilities. That's also true with carers of mentally disabled adults (for example, an adult with the mental age of a child). But it's still a crime if you whip those people with a belt for doing naughty things, like playing with the computer against your instructions, even if you use a looped belt or less damaging instrument.
So which is it? Are the reasons why it's illegal to lightly belt a wife or disabled adult not very good, or is there something magical about the person being your daughter that makes those reasons suddenly become irrelevant?