I would never take an unpaid internship, let alone one where I would have to incur living costs as well. If they aren’t paying you, it is likely not a great opportunity; either negotiate, or start applying to other firms.
Most startups I have interviewed with bring up the salary range early on in the process, and ask whether I am comfortable with that range before moving forward, so that it saves everybody’s time. The fact that it is an unpaid internship with a “performance bonus”, and the company was not up front about it from the start, makes it seem even less legitimate.
Don’t sell yourself short, and keep applying to more places. Try reaching out to alumni working in your area of interest. There are thousands of companies who are willing to pay bright students at top universities; don’t go work at a place that doesn’t.
Additionally:
“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.”
I don’t think your parents have a misinformed opinion here; in fact, I believe they have it exactly right. It seems ludicrous that you would take on a loss to work for someone else, and is something I wouldn’t do even for a “big, prestigious firm”.