Sixteen is much too young to be parenting a parent. But sometimes life isn’t fair, is it? I can tell you are a sensitive and responsible young person. You’re right that therapy would be helpful for all of you. But sometimes you have to start small. I suggest that you ask your school counselor if there is a teen support group in your school or locally that you could join. Another possibility is to attend a family support group of NAMI (National Association for the Mentally Ill) to help you better understand what your mom needs and what is realistic for you to do. The link for your area is here.
Once you have more support and once you have learned some more sophisticated skills, you will be in a better position to be helpful. Finally, you might look into the free services at The Center for Group Counseling in your city. I don’t personally know anything about the Center except what I read on the web so this isn’t an endorsement. But the information on their website is promising. Once you get started in therapy, you may be able to get your mother to join you. Hopefully, your father will eventually become part of it too.
By all means, tell you mother that you are seeking help for yourself. Secrets are rarely a good thing. Reassure her that you aren’t blaming her and that you have no intention of hurting her. Ask her to love you enough to support you in learning more about how to manage your own feelings without burdening her. You seem like the kind of person who could explain that in a loving way.
I very much hope that things improve for you. In only a couple of short years, you should be able to go off to school or get a decent job and think about having a place of your own. Let’s use the time between now and then to help you find more help for your mother so you’ll be able to become more independent without feeling like you are abandoning her.
I wish you well.