Thank you for the information.
It sounds like your mother and siblings may be taking advantage of your good nature and generosity. If you are the main caregiver for your father and they use that as a way to exclude you from family events and to talk about you behind your back, that can be considered emotional abuse. You not only deserve a thank you for what you do, you deserve to be treated better than you are treated now.
You can try a couple of things to deal with the situation. One, you can look into finding someone who can stay with your father when there are family events. Your local Area Agency on Aging may have someone who could stay voluntarily for low or no cost. You can also hire a caregiver to stay with him.
Two, you can refuse to assist your family when they wish to see your father, especially your mom. If she wants to know why, let her know that you feel you are not respected for what you contribute to the family.
Three, you can stop dealing with your family altogether. When someone is abusive to you and will not address it no matter what you do to work it out, then unfortunately you may need to protect yourself by distancing yourself. Do what you can for your father, then let your family know the rest is on them. If they will not help, try to see an attorney about taking over care for your father and then arrange for support and help from others. That way, you no longer need to deal with the family issues on top of being the main caretaker for your father.
You may also want to consider therapy to help you cope with what you are going through. You are under a lot of stress
and having support can help a lot.
I hope this has helped you,