Fortunately, my parents finally came round to my way of thinking and I am glad to say they are most supportive.

came 是动词 come 的过去式，come 的基本意思是"来、到达"。 round 在这里是副词，表示"转变方向"或"改变立场"的意思。 came round 是一个短语动词，意思是"改变主意"或"接受（之前不认同的观点）"，常用来描述经过劝说或时间推移后态度的转变。