My moral framework can be different from yours. Me the individual can come to the conclusion that something is immoral when the rest of the group doesn’t agree with me. And (at least for my own moral framework) I should take action accordingly.
So I don’t need a shared framework to make the claim that something is immoral (to me).
The second is that it isn’t very interesting to stop at “personal moral judgement”. You’re having a dialogue, right? So, if you want to have a dialogue, you must explain your moral reasoning. I don’t like your parent’s use of the term “moral framework”, because it does lends itself to a relativistic interpretation - though charitably, the parent need not be a relativist, and is merely acknowledging the different stances of various moral theories. But also charitably, if we lack sufficient common moral ground, the first task is to find that moral common ground before you can discuss something with two incommensurate views in play.
So I don’t need a shared framework to make the claim that something is immoral (to me).