if "Tai-Yang" is "太洋" ，your parents's explain is right ,maybe it's include the meaning "foreign flavour"?this is my opinion. i can't correspond the "An-Jie" to chinese word ,so i don't know it's meaning..

Hi. I agree with spysnake. In fact each Chinese character has its own meaning, if you randomly combine two in a name,it won't produce a phrase but only a general meaning.Owing to the fact that lots of charaters share the same pronunciations,I really have no idea what An-Jie would be?Perhaps "安杰"?But it does not exactly mean well-rounded &generous,I'm afraid... Tai-Yang could be 泰洋 too. *Oh,sorry,I didn't see your question is about Taiwan dialect.I know nothing about that.Maybe "An-Jie" truely has such a meaning in dialect like what you have said in your post.