Welcome back on the platform and thank you for sharing your situation with us.
In the past, the immigration offices' ("Ausländerbehörden") ways of dealing with the issuing of "Ausbildungsduldungen" differed sometimes significantly from each other. Since the beginning of year 2020, the conditions for the "Ausbildungsduldung" are stated in the section 60 c/residence act (§60c AufenthG). Here it says, that the "Ausbildungsduldung" has to be issued (legal right), if all requirements are fulfilled. If I understood correctly, in the case that single elements are not met (mostly referring to the identity proof), the issuing/granting will be at the discretion of the "Ausländerbehörde".
We discussed the requirements already here, please see i.e. this thread for more details.
Without knowing your personal situation well, if you meet the requirements, I assume that you don't need to worry much. The Ausländerbehörde knows, that your wife and child are still in the asylum procedure, right?
@manrhei-mbeon or @mbeon-Ruth, would you agree?
However, please feel free to get back to us with any further questions. You may also want to assess your situation closer in a migration counselling office or with one of our mbeon-experts via the App.
All the best,