If your parents don’t have the money or can’t pay, they don’t have the money and can’t pay.
How can you force the issue? I’m assuming you don’t have $240K lying in an account somewhere?
As parents, we try really hard to work and save money for our children’s college expenses. Many of us are shocked at how much the costs of college have increased. You have to understand that when we went to college, your parents included, we could work part-time, and pay our tuition and fees. Everything has changed. You need to change your expectation of what you think you deserve. Go to one of your safeties, then if you want to go to your private school, go as a grad student and pay for it yourself.