(note that privacy and GDPR issues apply differently for children)
> natural rights to their data which they would otherwise have
This is not a thing. Data has traditionally "belonged" to the entity doing the recording of the data.
Suppose she in later life becomes a Hollywood star and her school starts selling these recordings of her on the internet because, after all, her father has given them a permission to do this for fifty years ahead?
This actually made me chuckle a little. I genuinely have no idea if you're joking here because this sentence is ridiculous.