Parents are responsible for the safety of their children.
Literal
Parents [topic-は] own [possessive-の] children [possessive-の] safety [for-に] responsibility [subject-が] exists.
自分の (one's own) clarifies the possessive — 'their own children,' not someone else's. ~に責任がある is a set expression: 'to have responsibility for X.' This is a general-truth statement — 両親 means 'parents' as a category.