Attorney Sinclair provided an excellent answer. I'd only add to your question, a PC has the authority that the parties grant to him or her. So, if such a naming issue is in the PC contract you all signed, then yes they can.
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Child development experts speak of preserving the role of bio-parents and finding other names for steps and close friends. Although a child may on its own choose to call a step-father "Daddy," most frequently that naming starts with the mother. There is no law in the form or statutes or regulations on this subject, but courts in case law have acknowledged this very naming issue as evidence of parental alienation. Nor is there any requirement that all the children in one home call the adult male "Daddy."
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The earlier answer on parenting research is spot-on. That said, however, I question whether or not there is an solution to this issue. I say this because, basically, the child is going to call your SO whatever they wish to call them, and no "parenting consultant" or court can change that fact. And, frankly, if the baby-daddy and/or you want to constantly take this back to court over such small, impossible-to-resolve issues, you are in for a long additional 16 years until the child is an adult. My advice would be that this continued litigation and conflict will cause more harm to the child than any "parenting alienation" claim either of you can make. There is no statute controlling this kind of thing, and, even if there were, it would be totally impossible to enforce.
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