Since my father is bedridden, we take turns caring for him.
Literal
Father [subject-が] bedridden [because-ので] we [subject-が] alternating care is-watching [polite-います].
寝たきり means 'bedridden' — literally 'lying-down-only.' 交代で means 'taking turns / in shifts.' 面倒をみる is a set expression meaning 'to take care of / look after someone.'