Me [27F] with my sister's [32F] pregnancy and birth, our mother[55F] has ruined both and I need help on how to help her cope with post partum and dealing with our mom

This is my first post and kind of long, so appreciate you reading through!
My sister and I have have always had a rocky relationship with both our parents, but it has gotten extremely bad since we both graduated college. For context my sister, Jane, is 32 and I am 27. We both have been low contact or no contact over the past two years on and off. Our mom shows shows classic signs of narcissism like telling us we are putting her in an early grave, we don't love her, she envies other families on social media, jealous of my sister and I's relationship, and that we are always out to get our little sister (22F golden child). These fights usually end with screaming, sometimes physical altercations, or one of us leaving. This has been a life long issue, but really my sister's pregnancy and birth have been the point I think we can't handle anymore. To give some color to the story below, this isn't just a birth issue. She has caused drama throughout the whole pregnancy. A good example is that she was furious my sister and BIL told my parents at a restaurant that they were pregnant. Her comments were how dare they not tell us in our own home, and disrespectful it was of their privacy.
The current issue is that my sister just had her first baby last week. My sister wanted just her and the husband to be present for the birth, and everyone else to be in the waiting room. My mom wasn't accepting this so while my sister was in delivery, she stormed out crying to the bathroom. Once we were finally allowed into the delivery room, my dad stood by the door with a sour a expression. Then my mom walks in with an angry face, looks sideways at the baby, and in a flat tone "oh she is cute". Then refuses to hold her, and they both leave after an hour. The next day she just shows up at the hospital, again with a ****** off expression, refuses to hold the baby, and tells the grandmother "I know how special this is to you, and i will give you the right to hold her first" to make the point of how mad she is. After they leave, I go back to my sister's house to find a giant plant with pink bows. My parents went to my sister's house, without permission, and left in her bedroom (the most private place you can be). To add fuel to the fire, I originally called nmom and said we should go to a florist and get a bouquet. Her only comment was okay, but you better not leave out golden child. Now back to the huge plant...it is signed love, mom, Dad, golden child, and grandmothers. Good job on making a double whammy and causing emotional distress to both children! The real kicker is what my mom then posted a Facebook status that night. She said "Really Sad how people intentionally make others feel unwelcome, left out, and excluded" followed by another post "I am not allowed to post any info nor photos of my grand daughter" My husband I were finally allowed to hold her today thanks to my husband". The next day she posts a head shot of the golden child (younger sister) "Love her!!!! I am so blessed to have such a kind, caring, and loving daughter". After this she blocked Jane and I from social media. Both mom and dad have been no contact with my sister since Saturday. Mom ignored my birthday on 11/15, but sent sappy I love you and miss you texts all day 11/16. Also, on 11/15 my mother posted on Jane's birth announcement "so very heartbreaking as her grandparents that you have excluded us". She is playing psychological warfare now by alienating and blasting Jane on a public forum, and trying to being all loving me after ignoring my birthday to cause me pain. I know her end game is for me to tell Jane what she is texting me since Jane has called over 16 times, and neither parent has responded. This is a classic strategy she tries to get pit my sister and I against each other We are both at our wits end. And the saddest part of all is that Jane's MIL is with her, and is acting how a true loving grandmother would. The stark contrast between the two made the issue even more glaring and heartbreaking. We both have so much emotional damage that I am scared it will lead to my sister having depression while dealing with her newborn baby.
Any help is appreciated, and again so sorry for the long post.

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tl;dr: Mother ruined my sister's pregnancy and birth with her behavior, and I need advice on how to help my sister cope with this and dealing with her first baby


Source.

Im not sure how helpful this is but maybe you and your sister should distance yourselves from your parents. Ignore them in a sense. Maybe that will bring your mother around and make her see what she has put y'all through.