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I once was discussing a personal decision I had made with an acquaintance, and he expressed his disapproval of my decision with the phrase "It's not about you".

As far as I could understand, this phrase somehow means "You're acting selfishly and you don't have the right to do this". But I am having a hard time putting that into the words "it's not about you". On the surface the words are gibberish. Of course a decision I make for myself is "about me". Does the phrase mean "a personal decision that affects you should not concern you"? That also makes little sense.

Of particular concern is the issue that it's impossible to refute without understanding what it is trying to say. So the recipient of such a statement (i.e. me) has no choice but to accept this vague statement of disapproval without objection.

What is the correct way to understand how to parse this statement?

Your understanding is correct (about the general tone of the phrase).

The parsing is indeed a little harder, but boils down to something like “the narrative of this situation is not (just) about you”. Whether that means not about you at all or not just about you is vague.

'It's not about you' only states that the matter does not include oneself. But that does not exclude oneself from having the right to comment upon it, or to take action about it.

If somebody threatens my neighbour with a knife, it is not about me for I am neither doing the threatening nor am I being threatened. But it is my right - and, indeed, my duty - to inform the Police, go to the aid of the victim or do whatever else is appropriate.

Grammatically, 'it' is the subject of the sentence and 'you' is the indirect object. The sentence makes it clear that 'you' is neither the direct object nor even the indirect object of the matter in hand. It is the prepositional object as made clear by @Janus Bahs Jacquet in comment.

But this does not, necessarily, exclude me altogether.

It may not be 'about' me. But that does not, necessarily, mean that it does not concern me.

Putting it coarsely, the issue is about the “you” person being selfish. Here are three examples, ranging from strong to weak to false.

Strong

Dad has hurt his back badly. It is a great effort for him to get out of bed to go to the toilet. He could drive a car to save his life, but otherwise it is out of the question.
His daughter’s weekly ballet lesson starts in half an hour. She rants for ten minutes about how she likes ballet, and how much she needs to go regularly in order to do well. She launches into a speech about how little Dad loves her. Dad replies gently, “It’s not about you.”

The point here is that, although the ballet is important, Dad has a problem that if far more significant.

Weak

A group of friends is going out to celebrate the birthday of one of them. They end up in a restaurant that does not have gluten-free food. One of them comments quietly to another that he is gluten-intolerant (which means that he won’t be able to eat anything). His friend replies, “It is not about you.”

The idea here is that it is a kind and generous act for the gluten-intolerant guy to not disrupt the celebration.

False

Daphne rings up Robert; she wants to see a movie, and wants someone to go with her. Robert apologises; he has an major assignment due in two days. Daphne abuses him for being uncaring and anti-social. “It’s not about you!”, she yells.

The idea here is that Daphne is just too focused on herself to see that Robert really can not afford to come to the movie.

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The point of the expression, “It is not about you.”, is the implication that it is about somebody else (usually the speaker) — that what matters the most is some issue someone else has. As a rule, when someone says this, the “you” person does have a relevant issue; if the objection is correct, the point is that that issue is not as important as the issue the speaker has. Conversely, it can just mean that the “you” person is being selfish. (It is not that surprising if the speaker is actually being selfish (nor if the other party is being selfish and falsely believes that the speaker is being selfish).)

Note that, in the “weak” example, the idea is not that the issues involved are weak; the “you” person does have a legitimate objection, but the opposing issue — the birthday — is stronger.

I do not know the right terminology to label what it is that “it” refers to, as a grammatical concept. In everyday terms: it refers to the situation generally, and particularly to the issue that has given rise to the interpersonal conflict of interest [think “argument”].

Another explanation is when "It's not about you" is said to someone who is worried, concerned, or puzzled about someone's (negative) words, thoughts and behaviors towards them.
If a friend of the person who is puzzled or concerned says "it's not about you" to them it means it's the folks who are acting in a negative way have some kind of issue and not the recipient of their actions. has an issue, problem etc.
So, basically it's a way to let someone know that they're okay and it's the other people who aren't - for whatever reason that might be.