Parents have great i____ over their children.

impact 对……有影响
或influence可是这两个动词都与on搭配使用啊influence可以和Over连用，也可以和on 连用。 用influence 吧。我看牛津和英汉多功能上都有例句的。He has no influence over his children.这是外研社英汉多功能字典上的例句。