Your parents aren’t being realistic about affording any college on $16k without loans. I was that parent, didn’t want my kids to have loans, wanted them to pick colleges they could afford with just merit aid and my contribution. Alas, things don’t always work out that way. They have taken out loans, although less than the maximum. The world did not end with them taking the loans.
You need to look at all the options, the NY state ones, the merit ones, and then Cornell. I really doubt you’ll get everything you need to go to Cornell even with the loans and $16k from your parents, but see where your research leads you and be prepared to go somewhere else.
Coaches at a lot of schools are the highest paid state employee. There aren’t a lot of people standing on principle and not going to UCLA, Cal, or Michigan because the football coaches make too much money. You may not want to go to Alabama, but it may be the best deal for you. If the option is Alabama or nothing (you might not find any school you can go to for $16k), would you take ‘nothing’?
You seem to be arguing with the responders here, but they are just telling you their experiences. You might not be able to convince your father that Cornell is as good at HYPSM, but the group experience is that people do get jobs out of other schools. You have to deal with your father, and it might cost you opportunities because he’s only willing to pay for the best (and the reality is that you may not get in to those schools). Our group experience is that often loans are necessary. Our experience is that families with incomes above $200k do not normally get need based financial aid. You asked, those were the answers.