cashstasherat23, you might want to remove your dad as an authorised user right now, before you talk to your parents again about the brother's proposed loans. There is a risk that your parents may decide that if you are not going to cosign the loans, they will just take a large cash advance on your credit card to finance your brother's education themselves. They may even rationalise it as merely doing what you should have done. I wouldn't take the risk. I would contact the credit card company this very moment and remove your dad now before he even get a chance to consider how he could use the card to help your brother.
I'm going to reiterate Cathy's advice here. Your parents are not behaving rationally in this matter. If they are sitting on such a big pile of cash, then why don't THEY loan your brother the money from their own retirement accounts? Oh, that's right,
because that is a terrible idea! Just like asking your child to co-sign another child's student loan is a terrible idea. So, it makes me wonder what other irrational ideas or actions they will pursue . . .
You may think that your Dad (or Mom) would never do something like take a couple of $10K cash advances from your card (in this case for the "good reason" of helping your brother, of course) without asking or telling you about it, but I had a family member do exactly that to me once. Prior to it happening, I
never would have thought the person would do something like that. Thankfully in my case the second cash advance attempt triggered an "usually high cash advances" request flag and they called me because it tripped a fraud-control wire in their computer system. I immediately closed the account on the phone as the caller recommended, but there was nothing they could do about the first $10K already advanced since the advance went to an authorized user, which they don't consider fraud. I had to get a lawyer involved to make the other person held responsible for paying that money back.
So, take your Dad off the card. Just tell him that you are taking him off so he doesn't get embarrassed by a declined card, and suggest that a prepaid card (like a Serve card or Target Redcard) is probably his best bet for places where he needs to use credit cards. He doesn't need to be "building back up his credit" while he is in bankruptcy. Probably plenty of companies will offer him cards either way once enough time has passed. He just wants the convenience of having a credit card, and he is taking advantage of your kindness in the process. It sounds to me like both your parents and brother need to be on "cash only" for awhile since they've proven they can't handle credit responsibly.
Oh, and I'll third (fourth? fifth?) that your "No" should from this day ever forward be "We've already discussed this and I'm not changing my answer. I'm not going to discuss it with you anymore. Thank you for your consideration." You are under absolutely no obligation to call them to discuss is further.