"My dad has a lion."
Translation:Il mio papà ha un leone.
You don't use the article with the singular as long as you're talking about a close relative. So you'd always say, "mio papà," or, "mia mamma," or, "mia sorella". But if it's not a close relative, that's when you'd use the article: "il mio amico," or, "il mio insegnante". You always use the article when it's plural, no matter what: "i miei fratelli," or, "i miei genitori".
You only use the article if it's the word papà and babbo. For the word padre, we omit the article. It's a rule.
Ex. Il mio papà ha una mela. <-- My dad has an apple. Correct Il mio babbo ha una mela. <-- My dad has an apple. Correct. Il mio padre ha una mela. <-- Wrong. Should be: Mio padre ha una mela.
Same goes for madre and mamma.
I'm afraid I'm just totally confused, we seemed to learn a rule then suddenly ignore it! The point I would like to clarify is this. Regardless of which is 'correct', would both be understood? If the answer to that is 'yes', then I shall move on, happy and contented, to the next lesson along with all my mistakes!