as many things in many countries (not only Germany) at the end it comes mostly to the person you have in front. My family (wife, Son and me) are also Third country nationals, I do work as an Engineer, I mentioned this, because, the answer from government is always catchy.
Long story short, my wife did not get the money, (i did), the reason according to the Stadt Düsseldorf for my wife not to get the money, is that, she needs a work permit, as a requirement. I mentioned that I am an Engineer, since according to the law, my wife was supoused to get also a "work permit" in the visa since the begining.
So after 1 year of trying to convince auslanderamt that they made a mistake, and finally get the "work permit", to comply with the requierements for Erziehungsgeld (elterngeld for the wify!), well they told me no, and after I rise a complain, well they told me way no! and now I have to pay the trial expenses!
So, if your wife has a Work permit, (or is an EU national or German), then yes, you are on the safe side, if no, well worst thing is that you must pay trial expenses like me instead of getting any support