How do I parse 宋耀如一家用过早餐?
Sometimes, it helps to know the names and background of the people you're reading about. The appearance of the name 宋慶齡 in the text should instigate an immediate background check.

宋耀如 is 宋慶齡's father - his proper name is 宋嘉澍, and 宋耀如 is his courtesy name.


一個星期天，宋耀如一家用過早餐，準備到一位朋友家去。
One Sunday, the house (family) of 宋耀如 had breakfast and planned to go to a friend's house.

The friend should be primarily assumed to be that of 宋耀如, and by extension a family friend.

FYI: 用餐 is another way of saying 吃飯, if not a bit more old-fashioned.