"No, my dad is not busy today."
Translation:No, mi papá no está ocupado hoy.
In reality, it is not simply 'Father = padre, Dad = papá"
There are rigional differences that in some places, it is frowned upon to call your dad papá if you are a fully grown up while in some others, calling your dad papá is the norm even if you are 100 and your dad is 130. Also, in some places, you call your dad papá when you address him directly while you refer to him as padre if you are talking about him in front of others.
So, forget about the father = padre, dad = papá formula as it does not work that way. It's much more nuanced.
And yes, padre should be accepted in this question.
The postings I just read are quite helpful here, and I understand why "padre" was marked wrong. But I would like a more general understanding of formal/informal choices. Also, how wrong is "Padre?" To a native speaker, would it just sound quaint, or would it be a serious gaffe?