Your parents pay for the internet, and they are your legal guardian and are responsible for you.
It is entirely up to them if you can use the internet.
And the point of adding that phrase to adverts isn't because the website in question might be inappropriate, but because there are inappropriate websites on the internet.
It's very easy for young children to get lost looking for webpages and end up in places they shouldn't be.
Think how many fake websites there are that rely on people mis-typing legitimate website addresses.
These can take people to all sorts of places, whether researchographic or violent or related to shopping.
Which is another point, in that it's easy for young people to accidentally purchase things online.
How many people have eBay or Amazon accounts that are constantly logged in with shortcuts.
An inquisitive child might click on these and find themselves playing around on the website, inadvertently bidding or buying things.
This happens constantly on things like iPads and iPods, where game-apps for children have premium purchases available, and young children can spend lots of money quickly by accident. That can happen on computers too.
Or, children might download things that they shouldn't. It could be very easy for someone to download a potentially serious virus from clicking spammy advertising links or pop-ups.
Even experienced adults can have this problem.
Young children are not suited for using the internet entirely unsupervised.
Even some teenagers who aren't internet-savvy can have accidents.
Innocence and inquisitiveness can lead to some unfortunate problems online. And if a parent can prevent them, then it's best that they do. And since these adverts are aimed directly at children, then they have a responsibility to try and get parents involved instead of just leading children to use the internet on their own.