Well, the only parent that knew I wear diapers was my mom, as I wasn't incontinent when my dad was still alive. But my mom knew about it after about the first year I began having problems, but she never really said anything to me about it.
My wife is the one who is always telling me that my diaper is too obvious, or can be seen, and I need to change my pants or put on a longer shirt.
As far as asking me if I need anything, she doesn't do that. I take care of all of my laundry and supplies if I need them. I'll let her know if I need something and ask her to pick it up for me, since she works in a grocery store, ( the pharmacy), and it's just easier for us if she gets it. That saves me a trip into town for it.
As far as the "D" word, lol, we all call them diapers because that's what they are. I don't worry about walking around in just a diaper at home either, since my son also knows why I have to wear them. He's fine with it, he's still got a slight bed wetting problem and he wears a pull-up at night because of an under developed bladder and he is a sound sleeper. Neither one of us ever says anything to the other about it. It's just accepted as a condition that we don't have control over, so why make a big deal about it.
Looking at this from the other side of your question as a parent, we do ask him if he needs any pull-ups, and remind him that he needs to let us know if he's getting low on them. And actually, he wears goodnights, but had worn pull-ups at one point, and so we just call them pull-ups out of habit.
I hope that answers some of what you wanted to know.