Quote:- "...I don't recall ever reading parents referring to their children as 孩子们"
That's because under normal circumstances it is too formal to refer to or address your children as such. If you say or address your children as 孩子们, the 们, (which in Chinese could in some context connotes a certain formal societal status or ranking), would be like calling or addressing your children as "my progeny", unless of course the special circumstances or context calls for it. (for example in drafting a will)
So, in the Chinese context you could use 孩子们 when, for example making a formal speech or about to give some sagely, fatherly advice, like,
孩子们, 我有个好消息。你们大哥，聪明, 毕业为医生. 希望他的成就会鼓励我们家的孩子们努力读书。
"My children, I have good news. Your elder brother, 聪明, graduated as a doctor. Hope his achievements will encourage our family's children to study hard"
(Well, assuming this guy has 3 wives and 15 kids)