Did you keep the details of the Cornell NPC estimate? That could provide support for more aid (assuming the NPC inputs were accurate) when you talk with their FA staff. Are your parents divorced, own real estate beyond a primary home, or own a business? All of those things can lead to inaccurate NPC estimates, as can forgetting to include assets, trusts, adding 401k deductions back to income, etc.
If CalTech is affordable, it doesn’t seem Cornell is an option, assuming the $32K is the final number…how would your parents be able to close the gap from $32K to $20K (and you say the $20K is already a stretch)?
Do you have any other affordable options?