For the English sentence to be grammatically correct, the word "dad" should be capitalized. In a situation where words like "mom", "dad", "grandma", etc. are used as proper nouns, they should be capitalized, while in other cases they are not. The quick way to determine this is to try substituting the person's actual name instead; if it fits, then the word should be capitalized. Examples: "Invite Todd" works perfectly, so "Invite Dad" is correct. However, "Invite our Todd" doesn't make sense, so "Invite our dad" is the appropriate choice.
In the Spanish lessons, these sorts of simple present sentences usually accept either the English simple present or the present progressive. And usually the meaning is not very different (well, except for the fact that simple present in English tends to mean "habitual action," and present progressive means "at-this-very-moment"). The little bit of experience I've had with the Italian exercises seems to suggest that both simple present and present progressive work as translations, too.
However, for this sentence, the meaning difference would be quite substantial. "Why don't you invite dad?" sounds like a suggestion and makes no presupposition that the addressee is opposed to doing so, whereas "Why aren't you inviting dad?" sounds like the speaker believes the addressee has already decided not to invite dad, and the speaker wants to know the reason for this. Can the Italian "Perche non inviti papa?" be both of these interpretations, or only the first one?
Duolingo: you don't accept, "why do you not invite your Dad", but do accept, "Why don't you invite your Dad". Why do you not accept my answer, or will you only accept my question if I write, why don't you accept my answer?