Parents should be held personally responsible for their children's actions.

personally 是形容词 personal 的副词形式，由 personal（个人的）加上后缀 -ly 构成，表示"亲自地、个人地"。 responsible 是形容词，来自动词 respond（回应），意思是"有责任的、应负责的"。 personally responsible 表示"个人亲自承担责任"。