Shorter answer: The chances of you getting salary equivalent of benefits you reject are slim to none at best. Some companies out there might decide to give you a token amount in your paycheck if you opt out of benefits, but don’t expect that to make a significant dent in your overall compensation.
Benefits might be portrayed as some dollar equivalent bundled in with compensation by human resources (or a recruiter or someone else on staff) but the reality is benefits are not some magical coupon that can be exchanged for compensation just because you don’t need the benefits.
Longer answer: You say this; bold emphasis is mine:
One of the benefits that this start-up provides is a health care plan,
which I don't need because I'm covered by law under my parent's
insurance until I'm 26.
This “benefit” is not unique to the startup or any startup or any company for that matter that has full time staff. Health care (aka: health insurance) is a common benefit most any company offer full time—and some part-time staff—as part of a standard package offered to employees.
So this is not special.
That said, I believe I understand why you believe you can somehow negotiate the value of such a plan with your overall compensation: Many human resources departments—or people acting as such in a small company—will often tag a raw dollar “value” to benefits like this to make it seem like your compensation is tons more than it is.
So—for arguments sake—someone in the company will tell you that you are going to be hired at a salary of $90,000 (for example) but the benefits are “so amazing” that they add up to $20,000-$30,000 on top of your base compensation… So it’s like you are getting paid $110,000 to $120,000 and isn’t that “Amazing™!”
This is “truthiness” at it’s best: Yes, those benefits might add up to an equivalent of $20,000-$30,000 if you were to purchase and use similar services on your own, but—as you point out—you might not ever need those things so the net value is $0 to you.
Everyone knows that is the deal, but the nature of group benefits like this prevents any one-to-one equivalent in negotiating that discounted value with a raw dollar figure.
Meaning, utterly no company in the world will ever say, “Okay this new hire doesn’t need insurance so let’s just give them the dollar difference in their gross salary.” If anything you even asking a question like this might make the company think you are simply naive and that might negatively impact your ability to be hired by the company.
That said, even if your parents have coverage my advice to you is to only fall back to that if you have no choice. If you are an adult and are being offered a real job at a real company, it’s time to be a part of the real world and be independent.