You do understand that there are creeps out there grooming children, right? Parents definitely do need to have oversight over their own kids.
Children should absolutely not have privacy on the internet.
The ID requirement is terrible, but saying that children need privacy to explore their sexuality on the internet is very problematic.
If this is the position the UK government holds then that brings into question their desire to protect children online in the first place.
Sexuality is part of the human condition that doesn't start on one's 18th birthday just because over-protective parents want complete control of their child.
It's weird that parents believe they need to control every facet of their child's life, down to being able to learn about why they feel the way they do. It's a form of abuse and coercion.