Parents bear responsibility for their children's education.
Literal
Parents [topic-は] children [possessive-の] education [for-に] responsibility [object-を] bear.
責任を負う (to bear responsibility) is more formal than 責任がある (to have responsibility). 負う literally means 'to carry on one's back' — the metaphor of physically shouldering a burden. A general-truth statement about parental duty.