I am so far behind in all this, and every time I think I have a handle on the case, I see a comment here with a link to contradict what I thought was correct!
I did just watch the first video on this thread, of Skylar talking to her parents when they first found out about the baby. I thought her father handled it very well, seemed naturally shocked and aghast at the circumstances, and struggling to understand what was going on. Something that struck me was that he was very fixated on the fact that the doctor gave Skylar a prescription for birth control pills at the same appointment where she was told she was pregnant. The father correctly sees this is very strange and inappropriate, and keeps asking S about it, and she gives tearful evasive answers. He's concerned that she says she starts taking the pills straight away and explains to her that birth control pills are meant to trick your body into thinking you're pregnant, so absolutely inappropriate to take in pregnancy. His line of thinking seems to be that maybe taking the tablets harmed the baby.
I've read elsewhere on this thread/ news items, that Skylar was distressed at the pregnancy news and asked the doctor to give her the BC script so that her mother wouldn't get suspicious, as she wasn't ready to tell her mother she was pregnant; the doctor reluctantly agreed. She couldn't explain this away to her parents because she has to admit that she convinced the doctor to help her deceive them.
It seems to me that Skylar has a pattern of panicking and setting up an elaborate lie to cover her tracks and if you focus on one small specific lie - as the father did in regard to the BC prescription - you can lose perspective of the big picture. This might be why there are so many contradictory stories about what happened to Annabelle during birth and death. Skylar keeps shifting and rearranging her story to placate the last person she spoke to, but she's never actually addressed what really happened. As her high school writing teacher pointed out, she can write out the plot, but she can't address motivation.