It has been a few months since I last played the game but if I recall properly you need to ask Lily to be your girlfriend. That is what promps if one of the criteria are met for getting Jennifer's blessing.That's the thing. Her Love Story Progress lists the following things, 'Lily agreed to help you to test your serums.' 'Work on getting Jennifer to accept your relationship' Her Special Actions list only list 'Help her take Insta-pics' and 'Ask her to test serum' aside from the generic 'Follow
For work it also requires Jennifer's blessing for your lab doing so means she drops out of college. I do not recall the details of her stripper path but I do believe that she does not need permission or drop out of college for that.