I think this is being framed wrong, largely because of the way the admin communicates. But at its heart, deportation or removal is not a punishment or punitive proceeding. That is well established for decades, as is the principle that the standard is not the same as criminal proceedings.
Assuming things are as reported, here is the logic tree:
1. An administrative stay of removal is just that, administrative. It in no way implies a broader right to remain in the U.S. and does not foreshadow a favorable decision on the claim.
2. The father has clearly demonstrated that his claim to asylum is invalid and not in the interests of the United States.
3. Under the law, any dependents attached to his claim are decided in lines with the principal applicant.
4. People with denied asylum claims are either removed expeditiously or referred to an immigration court for removal.
To be very blunt: The argument any of these dependents has a right to stay is the one that needs to be made. Legally, the case is far more easily made that they should be removed.