Yes, in most cases the parent, not student, can borrow up to the COA of the college. The student borrows first, usually the Stafford loans, and then the parent borrows, either a Plus (government) loan or a private loan.
Some student may not have a parent willing to borrow, the parent may not qualify, or as stated above, the family decides it is just too much money. “No way” may be not willing even if there is a loan program available.