These are great ideas. I think the best messaging system could be determined by talking to parents at each hospital. Every context is different. Most parents could want Facebook, texting, emailing, or a 24 hour phone line.
I think the usability of Facebook groups depends on the number of people in each group. Another way to limit spam is to set up some guidelines for the group, so group members know what the group is supposed to be used for. You could also create very specific groups, so that members are invested in those groups and limits the number of people in them. For instance, an "Allentown Fathers of November 2017" group, or something similar.
I have not talked to potential users, but that will always be my first step in any of my research or ideas. I put the idea on here the moment I thought of it. The OpenIDEO community will help me refine the idea.
The under-resourced mom, healthcare provider, and support systems could all be affected by this idea.