In the phrase:
As a father, ...
what are the functions of the "as a" part and the "father" part?
It would be more natural to split this [[As] [a father]]. This is a prepositional phrase.
"As" is a preposition. In this context, it means "In the role of".
"a father" is a noun phrase functioning as the object of the preposition. It is headed by the noun "father" with the determiner "a" (indicating a single undetermined father)