V.,
This is such a hard thing to deal with, for all of you! Your sister's anxiety is very real, whether or not it makes sense to your dad. On the other hand, it sounds like he must feel completely overwhelmed, and your mom is just stuck in the middle of all this. Your whole family needs help to learn how to deal with this, because anytime a member develops a problem, if affects everyone.
I sincerely hope your mom doesn't leave your dad. That will only cause more problems than it solves. It could also lead to more anxiety for your sister, as well as new problems for the other siblings. The best thing your parents could demonstrate to the kids is that when we have problems, we don't act ugly (like dad) or leave (like mom). We stay together and work together to find out how to deal with it.
Your sister needs a boundary set with school, even with this anxiety. She needs to know that, in order to learn to cope with her problem rather than let it become an excuse not to deal with life, she HAS to go to school! She will get support there...if it really is too hard, encourage your parents to look into alternate schools or private schools.
I think the other issue is, it gets expensive to deal well with these new sets of challenges. There should be counseling agencies that either offer help at no charge or on a sliding scale...call around. There also may be a church school or something that would offer a scholarship and be the right kind of setting for your sister. Bottom line, it will take some hard work and effort to find the right answers for everyone. But even though rolling eyes and leaving seems like an easier answer right now, it will only create much harder issues in the long term. Encourage your mom to think of all the kids and realize she would be better off finding a way to engage dad in the process rather than leave. I wish all of you the best; this is such a difficult thing for all of you...and especially your dear mom. God bless you!