The word responsibility can be confusing because its meaning changes depending on the "active worldview" of the person using it.
The old worldview is one of scarcity and competition, where one seeks empowerment through controlling people and conditions, and by being "right." In that worldview, responsibility means obligation, duty, and/or blame.
The new worldview is one of abundance and creativity, where empowerment comes from within and is expressed through partnership. In that worldview, responsibility is the ability to respond creatively, and it's an acknowledgment that each individual creates his or her own experience of life.
Today, pay close attention to how you feel when you're trying to be a "responsible" parent. If taking responsibility makes you feel heavy, burdened, guilty, ashamed, or resentful, then it's time to upgrade your worldview!