As the questions on here are to do with problems parents are having while parenting their children, should there be an option for asking and answering questions anonymously? Isn't it unfair to our children to detail their problems on a public website?
You can already contribute anonymously, although it's so obvious that it's in fact invisible:
Just log out. You can ask and answer questions without being logged in. When you do that, you'll be posting as an anonymous user.
Update: We have to assume that users have a certain level of ability to think for themselves. Posting things on the public Internet (especially with the content's open license) is, well, public, and we're not tasked with teaching people how to post wisely.
IMHO, whatever the parents might post is nothing compared to the compromising stuff the kids themselves thoughtlessly post on Facebook et al.