I have just come across the phrase "nachas rights" and Google does not lead me to a definition. It is in the writing of a British Jew, I expect from Stamford Hill.
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"Nachas" is a Yiddish word/concept that signifies pride in someone else's accomplishments, usually those of one's children. So by extension, "nachas rights" signifies the right of a parent to brag about such accomplishments.