Should I name a number first, or wait until they do?
Well, we had a deal with our kids before they started college on what they would have to pay for. Books, spending money, and any expenses related to an unpaid internship. So I get where your parents are coming from. There is no reason they should fund this for you. It might be a different story if you were a junior AND were in a field where unpaid internships are the norm (political science, for example). My advice is to go home and work this summer, and work part time during the school year as well. Get some money in the bank, then next summer you are positioned to take an unpaid internship and cover the expenses yourself if that is the best thing that comes up.
Yes, housing stipend (you’d need at least $1,200 for monthly rent) + commission would be okay.
DO ask around town where you live right now for similar internships/jobs.
Your parents shouldn’t have to pay for your internship. If you didn’t work during the year and save to pay for it youself, remember this as a lesson in independence and put it to use next year. Hopefully you’ll get a paid internship offer throuhgh your career center, but in case you don’t you’ll have your own $3,000 and you’ll be able to take the internship. 
Directly to the point of legality, if you provide value to a company, you need to be paid. Period. It has nothing to do with whether you can afford it or whether they will rescind the offer. If you applied to a job that offered you $1.50 an hour, you would not legally be allowed to accept those terms. If, in this case, you are offered $0.00 an hour with an undefined commission, you are still not legally allowed to accept those terms.
On the other side, if the commission is paid based on a defined set of parameters and backed by a minimum wage, it’s legal, and the only other question is whether you can afford to live on that. For a little while before my senior year, I had an “internship” with no pay except commission, and I was so desperate to have an internship that I accepted it. Now, I made decent money, but it was by no means an internship - it was a straight sales job and, after some further research, it was even listed as such on job posting websites! Same role, same company, different description. After realizing I was getting no legitimate experience, I quit and went back to waiting tables.
If a company refuses to pay you for providing value, then it is not a good opportunity. There is no ambiguity here. If you fear an offer being rescinded because you ask for the company to honor your legal rights, there is a problem here. The term “internship” is completely irrelevant unless you truly are adding absolutely no value whatsoever and the company is contributing to your education in a meaningful way (“experience” doesn’t cut it). If you do a single task that could add value to the company, then you are an employee who needs to be paid.
Also, if you are terminated or your offer is rescinded because you requested the company honor your legal rights, you have the right to sue for damages, which would give you plenty of experience in an area you never even thought of! hehe
My parents refused to pay for me to do an unpaid internship in another city. They sent me this article as a point to back up their argument:
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”.