This is a Ted Ed vid that briefly summarizes the main benefits of learning to play music that anyone would want to point to. This vid has literally nothing to do with Islam, but it may satisfy the curiosity of anyone who wants to know what sort of measurable benefits are associated with learning to play music.
https://www.youtube.com/watch?v=R0JKCYZ8hng
Although it's worth pointing out that a basic, intro-level high school music class is only going to introduce people to the possibilities of truly learning to play an instrument- it's basically an entire course whose purpose is to position a student at the edge of something and find out if they're into it.
It's too bad that you haven't been able to get anything to go your way yet, although I can't say I'm entirely surprised by the requirement for parental involvement in the exemption. That's probably done in order to prevent lazy students from opting out of something that they just don't feel like doing- which is not you, that's not what you're doing at all, but the requirement is probably there with that sort of thing in mind. I'd still recommend talking things over with your imam, your dad clearly doesn't trust this site so you should get your dad talking to him.
I do encourage you to continue to engage, to make a reasoned and well-supported argument and to take whatever steps you have at your disposal to try and make some progress. Please don't react with anger though, this is the exact sort of time and situation in which open displays of anger and frustration will immediately and predictably be interpreted by your parents as signs of extremism and cause for concern.
Again, do talk to your imam with your dad, and it might be a good idea to make a point of checking in with him on an ongoing basis, when you happen to see him and as long as this is going on. As things continue to progress, it may be the case that you have to compromise on something in a way that you didn't really want to. Of course it's not what you wanted, but it may be that you have to. So you'll need a particular type of guidance in terms of negotiating a non-ideal situation.
From your dad, it sounds like the extent of this guidance will be "
You have to do this, now deal with it." But....how exactly, Islamically speaking?
It well may be that the guidance you get from this forum is "
You shouldn't have to do that. You really shouldn't. That's wrong, you shouldn't have to." But....you do. So what now?
That's where your imam comes in. Even if he's not able to help you reverse direction- which you can ask him about, it's worth a shot- he's definitely going to be your best shot at having a neutral observer, who knows Islamic teaching quite well, who will most likely hear you out without judging you for being frustrated, and will help you find the most proper way forward even if you aren't able to get out of the class.
If I were to hazard a guess, I would say your imam can probably come up with a better course of action than faking sickness. I'm not sure exactly what he would want to say to you, but I cannot imagine that he would tell you go ahead and fake sickness, that's the right idea, do exactly that. I don't see that happening.