We don't know for sure.. But typically the child 'gets it' from the mother in very early child development. There is both a genetic aspect (which you may not have received) as well as the nurturing aspect. Not to say your mother is 'bad' or didn't do 'well' when your sister was young, but really that a BPD mom doing 'the best' she can while untreated simply have certain emotional deficiencies due to her life.
That said, it isn't impossible for an attachment disorder or dysregulated emotions 'from' the father and abuse or crises at a young age (not related to the mother).
However, your mother's current internalization and codependency is dangerous for her no matter what AND it isn't unlike many of the 'lash inward' BPD's I've read about. Your sister reminds me more of my 'lash outward' wife.
No matter what, your mother truly needs therapy. She needs to get stronger and more balanced, she needs to be allowed feelings and control in her life, she needs to stop being abused, she needs to grow-- and I hope when she does she leaves your father and sister.
Your sister needs your mom gone. She needs to stop using her as her whipping boy woman and having that validated by your father. Your sister still won't get any better until your father stops enabling her. She needs to hit her bottom and have no one there to take the blame or save her if she will EVER desire change enough to blame herself and step out of denial to work on it.
-So sorry you are going through this-
I have a BPD diagnosed sister, a mom who 'internalizes' everything (used to blame me for her emotions.. if I was doing things unacceptable to her then she wouldn't eat or sleep and my parents would both 'blame' me for my mom's behavior in response), and I have a dad that is definitely a micromanager and control freak, with a few relatively benign NPD traits.
I'm not in the same situation. It was very difficult for me as a teen, my sister was bad off, my dad an alcoholic (recovered now), my mom starving herself and not sleeping, always in turmoil 'because of' me or my sister. I was the 'golden child' meant to rescue everyone and keep the peace. However, I can honestly say that my sister wasn't acting out as extreme as yours (she is half and half lash-out or in or more lash-inward), my dad is controlling and sometimes unfeeling to others, but always has clarity afterward and taught me to always apologize and make amends and a caregiver if not a caretaker.. he is really a great dad in the end and his strength and compassion really helps my mom and sister and myself keep it together.
I guess there really are a lot of parallels between the nuclear families we came from, especially when I was younger like you and things were in ruin (I'm 31 now). I hope and know that your family can perhaps get through this, but honestly I'm mostly just concerned for you and your mom. If your sister and dad can't get healthy you two will need to get therapy and hopefully either get out of there or when she is willing and able I hope your father makes drastic healthy change in that home if he desires her not to leave him.