It looks off topic, but it's not. Here's a summary.
mykwyner replied to you, 'It is very common (and natural) for us to say, "These are them," "That is her," and "It is us," but they are not correct.' To which riverkid responded, 'Natural and common determines what is correct in grammar.'
In other words, mykwyner tells us that what people say although natural and common isn't always considered grammatically correct, whereas riverkid adds that if speakers are using "me", that if "me" is natural and common, then it is grammatically correct. After all, speakers use rules and rules are the grammar. Now, here we have two posters who disagree on whether "me" is grammatically correct or not, and that's your question, right? So, to that debate, dihen offers a truce with this.
'...formal grammar and informal/non-standard grammar are like two different sets of grammar.'And dihen is right. Why debate "me" or "I" when there are two choices available?
In short, kohyoongliat, as an predicate nominal use either "my parents and I" or "my parents and me". The latter won't make you sound like a non-native speaker; the former might make you sound pedantic. In the end, it all depends on who your audience is at the time. As for writing, especially formal English, what do your role models do? What do your teacher think? Who's your audience? That's a good question to ask yourself.
I use both.
All the best.