negotiating internship salary (urgent)

Okay to be honest, the main problem I have is not necessarily the living costs so much as my parents don’t believe there is an incentive to me taking an internship summer after freshman year. I personally am grateful for this unpaid opportunity. From what I know, I don’t think spending ~$3000 is extremely financially troubling for my family. They just don’t believe that the $3000 is worth investing in for an internship that isn’t at a big, prestigious firm–an opinion that is misinformed and biased, I am aware. They just really want me home for the summer. So I’m trying to make the situation so that there is nothing to lose by taking this internship, hence asking for a base salary.

So do you think that taking this internship unpaid is worth it, and I should try to convince my parents once more, or do you think I should risk them rescinding my offer by asking for a base salary?