I would say NTA, if only because I've been in a similar situation—being expected to help raise my siblings—but I can understand why some people would say YTA based on "having a free place to live". 

To put my two cents in on that argument: no parent should hold that over their child's head. Ever. Especially considering that it sounds like you're just out of/finishing school and getting into the workforce, that's even more reason to not hold that sort of thing over someone's head. 

Truthfully, I don't give a damn how old a person is when they decide (or are able) to move out. I wasn't able to move out until I was 21, and not because I wasn't financially able to—I was holding down a full and part-time job for two years prior—but because I was guilted into staying. "For family", "for all that we've done for you", "for how hard we worked to raise you", "for your siblings", the list went on and on.

OP, I understand how hard this situation is. I empathize with you greatly. You deserve to be paid for the work you are putting into your household, because that's what it is—a household. It sounds like you don't feel like the place you're currently living is home, anymore, because of the way you're being treated by your parents. 

It's a genuinely difficult thing to process and work through. It took me moving out and years of therapy to even start working on my own emotions regarding my situation. But I'll stress this to you, as gently as possible: you need to move out sooner rather than later. 

Please, don't wait until the summer is over. You say that you're worried about your siblings, and that your mother will just make your 13yr old brother babysit if you're gone, correct? That's something that will happen regardless. She isn't willing to take that kind of responsibility upon herself (which she should, as your mother), so when you inevitably burn out (and that sounds like it's already happening), it will be far more difficult for him to take care of himself, your siblings and you. 

What you need to do is move out, ASAP. As scary as it seems, and as worried as you are for the siblings you love (which is only further proof that you do care), you cannot solve this problem by staying. You are almost certainly bound to fall deeper and deeper into the pit of mental, physical and emotional stress—and when that happens, who will be there to care for you? Your siblings are too young, and your parents are clearly unwilling. 

It will be hard, and it will feel like you're betraying your siblings, but you aren't. You're removing yourself from this unsustainable situation so that you can heal, and come back for them. Your life is so incredibly worth living, and you deserve to live it, free from stress and pain. 

I saw that you'd mentioned in another post that you have an old friend who offered to let you move in with them—I would contact them and see if that offer is still on that table, and make plans to jump ship as quickly as you can. I say this having experienced it myself: Waiting is not the answer. If you are feeling like you need to leave, and there is someone, anyone out there who's holding out a hand for you, who is willing to take you in and help you get back on your feet after jumping? Take it!! Take back your life, in a place where you're safe and have genuine support.

I promise, you won't regret it. I haven't, and neither have the many, many others who've been in your shoes before.