I would not worry about his parents or his relationship with his parents. If he doesn't want to rock the boat, wants to do things their way while he's with them and live with you when he's not, etc. and you are okay with that, then fine.

I believe your question was about how you were behaving to him, and whether you were being rude to "invite yourself" along with his brother. I don't think telling someone else what we think and feel, when it is about us is ever rude. You gave him information, that you wanted to go to the beach with him. He didn't know that until you asked. You are not rude for asking and his response "sure, I'll pick you up" was a fine one. If he'd said, "no" that would have been a different matter, might have impacted your relationship differently.

But you cannot change him or his parents and their relationship. Only he can do that. All you can do is see if you mind the way it is, taking backseat. As we have heard, not everyone would "put up with" that but you may enjoy moving, having a new, good school to go to and living with him again. Leave how he relates to his parents to worry about if he ever asks you to marry him and wants to elope or do it in secret :-) Right now you're just going to school and living with a guy you love; nothing wrong with that! Good luck.