Parents have a responsibility for their children's education.
Literal
Parents [topic-は] children [possessive-の] education [toward-に対して] responsibility [subject-が] exists.
~に対して (toward/regarding) is the adverbial form of に対する. 責任がある uses an existential pattern — 'a responsibility exists toward X' — which gives a neutral statement of fact. The alternative 責任を負う ('to bear responsibility') would put more weight on the burden being shouldered.