I have had D since 1984 and also have a 14 year old, with whom I have occasional differences of opinion about various issues. Can you prove that your being in control is better for you than your parents? There's lots of threads that question the value of A1C tests but, as a measurement of long-term control to answer a question about whose control is better, I think that the test would provide a useful result. If your number is better than your parents, I would agree with your assertion that you should be in charge. Even if your number ISN'T better, I'd also say that you should be in charge but that they should perhaps have more input into decisions. Their input shouldn't be "don't eat that" it should be "let's figure out HOW to eat that?" and, if the first try doesn't work, try it again and figure out how to improve the result. As far as a logbook goes, I logged for a month to get a pump and that's all of the logging I've ever done. I like using the pump because the software keeps track of everything most of the stuff I need to do ok and its it into handy charts. It's less work for me than writing stuff down, although I have to admit I wouldn't do that anyway. That being said, if you have a person goal of "doing better" in any way, including proving that you are right and will do a better job controlling diabetes than your parents, the only way to do it will be to, in fact, do better.
I agree with CaryJ that perhaps your parents have an alterior motive in getting you to the class, not so much for D skills but to meet other people who aren't your parents who might help you. You are, of course, meeting us here @ Tu as well and this can be a good place to do research. Some of the people (e.g. Jen, who has been blogging very precise and detailed results about her adventures with swimming and D for a while now...), can get very useful information that may help them make decisions that will help their management improve. In some ways, another thing that helps me is not to focus on the numbers as numbers but to look at how much work is involved. I'm a busy guy, work, take junior to dancing lessons, cook most of our food (Mrs has hellacious commute, plus then I can be a food nazi..."look, grilled meat again!") and try to work out as much as I can. If my BG is out of whack and I have to wait around for correction boluses to get going or wait for snacks to work before I work out, I would be annoyed so I've sort of set it up so I do ok on my numbers. By doing "more work" to think about stuff, I am sort of arranged to do "less work" cleaning up messes. That being said, it is extremely difficult for people with diabetes to have a mess-free existence and even a "bad" number (low or high...) can still provide good data that you can use but you have to have a system of some sort, even a flaky, nebulous system like mine, to work to improve your results. I have to suspect that your parents are wanting you to kick some ass at diabetes as they will feel better about it at the same time you can feel better about it. While taking care of your diabetes is your job, it's up to you to prove that you can do it!
If they make you go to class, you should try a "test" and ask a question there and ask a question here and see how the questions are answered. I don't like a lot of the reports people give about how their doctors/CDE answer questions and, if you can prove to your parents that you can get a better answer here, perhaps they'll agree to let you bail out of the class, unless it costs money or something?