The parent is completely correct, why are you fighting it? This isn’t something you can out-logic, the words have definitions. Sex is the biological part and gender is the social & cultural aspects of sex.
According to Merriam-Webster, gender /includes/ the social & cultural aspects of sex. The exclusion of sex from gender is an extremely recent development that isn't widely accepted yet.
Also, motherhood and fatherhood are biologically gender-linked roles. They are measurable biological phenomena that tie strongly into your definition of gender.
Gender doesn’t exclude sex. Gender relates to sex, but it is not the same thing.
> The exclusion of sex from gender is an extremely recent development that isn’t widely accepted yet.
What are you talking about? The use of gender as a sex “role” was coined in 1955, over 60 years ago. Before that “gender” referred to grammatical gender, not to people. So if you use “gender” relating to people, that form of the word has always meant the social aspects of sex, not the biological aspects. https://en.m.wikipedia.org/wiki/Gender
There was never a time when “gender” meant the same thing as “sex”.
> Also, motherhood and fatherhood are biologically gender-linked roles.
You’re confusing yourself. Being a mother & father are sex based facts. Fatherhood and motherhood as words that can mean that someone is factually a mother or father, or it can in context be referring to the gender roles of motherhood and fatherhood. The stereotype of a father playing catch with a son is a gender role, not biology. You’re choosing words which have both meanings, which doesn’t help you understand what sex and gender actually mean.