No. Parents are not responsible for the Federal Direct Subsidized/Unsubsidized Loans. Students do not need to have a cosigner on those loans either, even if they are under the age of 18 because the “defense of infancy” does not apply to federal student loans.
If a parent is responsible for a loan, it would be the Federal Parent PLUS loan. Not all parents will qualify for the PLUS loan. As mentioned above, generally the student and the student alone is responsible for repaying any and all of their educational loans.