Not necessarily.

Recently we got a new LG TV that has an age lock option with some other family settings. The parent can turn it on with a PIN, and they can set up restrictions.
The same approach could be used here. But this would need 2 things: obviously support by the web browser app, and support by the OS to tie app installs, uninstalls and data wipes to the parent's code.

To help with cases when the device is sold, maybe the parent should press a button every year that they still want this, and also receive an email notification when the period is nearing it's end.

But a much better solution is that parents are dealing with their children.
They may see up some site filters, but when they notice that their child is using a workaround then it should be punished with taking away the phone.