"My father is the director."
For some possessive+noun compounds where it's a given there is only one, articles aren't used; "mio padre" is one of those, while for some reason you could say either "il mio papà" or "mio papà" (my dad). Same for "mia madre" and "la mia mamma"/"mia mamma" (my mom). Interestingly, when speaking about some other person, you could instead omit the possessive: "Il padre è avvocato" equals "Suo padre è avvocato" (his/her father is a lawyer). I'm afraid I can't point you to any rule though.
From what I understood, the article is omitted for all singular (!!! careful!!) family names (madre, padre, zio etc) unless they are "affectionate" such as "il mio nonnino,il mio fratellino, la mia zietta etc". "mamma, papà and nonno" are part of both groups so the use of the article is optional. I am from northern Italy and I've never heard anybody say "il mio nonno", only people from Tuscany. But since Italian is Tuscan, then I guess it's ok to say.