They want to fulfill their responsibilities as a parent so that they can take the load off their shoulders.
fulfill responsibilities=「責任を果たす」
SV so that S’ can V’=「S’がV’できるようSはVする」
Take the load off one’s shoulders=「肩の荷を下ろす」
They want to fulfill their responsibilities as a parent so that they can take the load off their shoulders.
「肩の荷を下ろせるように、彼らは親としての責任を果たしたいのです」
ご参考まで