It's illegal to allow minors to view research, whether it is being provided for free or sold. It isn't illegal for minors to view it or buy it.
If a minor views research and asserts that they are over 18, they've commit perjury. The webmaster has done enough to keep them from accessing their content, so their parents can't take them to court.
You can't specifically arrest a minor for viewing research, although you could arrest them for lying about their age under oath. They have commit perjury, which means they can face up to 5 years in jail if convicted. I don't think it has ever actually been done, courts have more important cases to try, the police have more important people to arrest, and in the end it almost always gets left to the family to prescribe a fair punishment. You wouldn't get a criminal record for it.
This is also true if you are under 13 and register on a forum without having your parents sign a waiver.