Old-fashioned grammarians would insist on subject case as the complement of the verb "be" (which only makes a difference for pronouns rather than nouns), therefore they would like to see the first. However, natural English calls for object pronouns, and so the second is more colloquial. Sometimes people get confused with pronouns when there are multiple people involved. A good way of sorting it out is to simplify just to one individual and see what you would say there. In this case that leads to either "It's I" or "It's me", and nobody I know would say the former on the phone.

You could use the second sentence and would probably hear it in non-standard English," They are my mother, my father and me." However in standard English, you would want to use a reflexive pronoun instead of me or I, for example, "They are my mother, my father and myself."

If you reduce the sentence to something grammatically similar, the answer should become evident: - They are ... me => It is me = me am it => ugh, that's wrong - They are ... I => It is I = I am it => that's correct!

....and I.