1. 当然是of,即：a new understanding of my father.
2. 这是一个比较固定的搭配：a/an/one's (...) understanding of sb/sth“一个对某人/某物的(...的)理解”.
再如：His understanding of English is good.
他对英语的理解力很好.
We must have a clear understanding of the problem.
我们对这一问题必须有一个清醒的认识.
A teacher must have an understanding of children.
教师必须理解孩子.