Here is another thought no one has mentioned. This adult teen mother most likely had prenatal care. She was under the care of an OBGYN. She likely had her child in a hospital and was observed by doctors, nurses, CNA's and various medical staff. She has likely had routine postpardom visits. She has a young infant 3 months of age. She has likely taken the child for routine well being visits which are frequent at this childs age.
My same thoughts, exactly!
If this is a typical example of how they normally treated their daughter when they suspected she was up to no good, I'd say they had significant part in driving her away. It is normal for children that age to be stepping away from the family and trying to find their own place in an adult world. I don't think they were ready to handle that in a healthy way.
Not that the priest is at all blameless. He should know enough to avoid even the appearance of evil.