I can understand your parents being upset.
When I was in labour with my first, DH mother came to see us about 2.30am, after having been out for the evening so was slightly worse for wear. She did ask if she could be there for the birth, but I told DH, no, so he told her he thought it was best to go home, which she did. But once my daughter was born the later on in the day, everyone came in to see us, my family, his family and they all had to take turns in holding her. She wasn't the first on either side, but everyone wants to see a newborn baby and hold them, everyone is excited. I was acutually glad to see everyone, tired but very happy and glad.
I hope it does get easier for your parents and your sil becomes more reasonable, because the first grandchild is very special.