The parent is free to ignore their identity (or so the comment implies). People who are LGBTQ+, black, female (in many situations), immigrants, etc. are not - those identities are imposed on them constantly by the people and society around them. They can't walk down the street, go to work, turn on a movie or the news, or even watch porn without people defining them by stereotypes. Their lives are often defined by it - their safety, their career options, their wealth, who will marry them, etc. The people who actually experience it keep telling you that - they can't ignore it, it's oppressive, brutal - why don't you believe them? Who are you to know better?
The obsession with identity is not in its subject, but in the majority observers.
> Obsession with identity is narcissistic by its very nature.
Ironically, actual narcisissm isn't focused on oneself as much as focus on everyone else - everyone else in the family, the organization, etc. has to be exactly what the narcissist wants them to be, to make the world exactly what the narcissist needs to feel safe.
That is similar to the identity situation we are discussing: The narcisissm is in the observer, who insists that the subjects should behave and act as the observer wishes.