And what about a required course in high school?
Further, minors cannot enter into contracts. Would Turnitin's license agreement be a contract, and hence void since one of the parties was a minor?
I know of no university that explicitly prohibits students from selling papers to paper mills (although I haven't checked). Prohibiting students from using papers from paper mills is a different thing entirely.
Rather they'd say they want your paper so they can see your thoughts. That's fair because they could get the same effect from merely talking to you about your paper.
Or they'd say that its okay to use outside sources (such as your paper) so long as they cite it. This is also true in many cases.
In that case, you're not breaking any honor code although you're aiding others in being dishonorable to their own codes.
Thanks to a quick google, here's my old student handbook: "Selling academic assignments. No person shall sell or offer for sale to any person enrolled at the University at Buffalo any academic assignment, or any inappropriate assistance in the preparation, research, or writing of any assignment, which the seller knows, or has reason to believe, is intended for submission in fulfillment of any course or academic program requirement."
It looks like my understanding of the honour code is therefore definitely false for my university, and, from the evidence, probably for many others. (This is particularly embarrassing because I was part of the group that approved the final version of the code!)
If the helper student has already passed the class, the professor can't take disciplinary action against them without making a major case about it. It's easy to simply fail the cheating student or give them a zero on the assignment without making a record.
Additionally, if it is brought to a committee, it'd be much harder to prove.
So although it's officially not allowed, there's not much universities are doing to combat the providers.
The university has the right to ask you to submit your work however they want as a condition to course completion. If you don't want to submit your work to turnitin you don't have to complete the course. Nobody's forcing you to.
If the extent of your objection is that turnitin robs you of potential financial benefit of selling your work, then that's by design: that's exactly what the educational institution wants to do! It doesn't matter that selling your work to a paper mill is legal. It's against the interests of the institution, so they're not going to let you go through their programme and keep that right.