When talking to others, when do you refer to your father as 아버님?

ANSWER: After he dies. Until then, you should refer to him as 아버지, and use 아버님 to refer to other people's fathers. You should, of course, use 어머니 and 어머님 in the same way.

"우리 아버님 살아계셨을 때, . . ."
"When my father was alive, . . ."