I disagree... it is wholly her business to share with whoever she would like to- even against her parent's wishes. Once they told her a truth about where she came from, it is unfair to put restrictions on what she does with that information.
How must that feel- to have a secret about yourself that you want to share but "can't" out of obligation? Who can she share with? If she can't be honest with those closest to her, how strong are those relationships? There will always be something holding her back from being truly herself...
Also, it's understandable that she wants to know who her biological father is REGARDLESS of her childhood and home life- learning about her DNA does not negate her lived experience.
Personally, I think her parents need to do some heart-work... They are not worried about how this affects their child (adult or not), instead they are worried about how it could affect them. They are essentially telling her that she can't be open and honest with anyone- which apparently includes her closest friends. Does that include the person who she wants to partner and spend her life with? I think it's wrong to expect or ask your child to keep a secret (dishonesty by omission) that could affect their entire life and future relationships.
If nothing else... what if the author's biological father (and thus, the author) has genetics that, due to her father's Jewish ancestry, put her future children's lives at risk? There are many diseases that are known to affect groups of people more then others, and that includes Jewish people... the author could be a carrier of such a disease and unknowingly pass it on.