Dennis Prager often says, "I prefer clarity over agreement", meaning that when he doesn't see eye to eye with someone else, and the potential for agreement has been explored and discharged, the most important thing is not convincing the other person that he is right, but rather having a conversation the nature of which the argument is made clear. He says this in the context of his radio show, where there are listeners of whom he considers it his mission to educate - clarity of position being the best vehicle for education.
At the end if the day, all we can do is make ourselves as clear as possible, hoping for agreement, but settling for clarity when it's the best we'll get. If someone shuts that down (like my father), I've done what I can, and he makes it known that he doesn't like where the conversation's going. It may be a personality trait, a manipulative tactic, or a completely subconscious coping mechanism, I don't know. But regardless, at that point I have a decision to make. Is it more important for me to be right, or to be clear? I've already been clear, so being right would require me to take the conversation a step further into territory where he's dismissively signaled he doesn't want to go. I can have tough conversations with my dad, and I do. But at some point, for the sake of relationship, I need to decide whether I need for him to tell me I'm right.
That seems one-sided, and to a degree it is. Because he cuts off a line of open communication, I'm the one who has to decide whether to push it, and if I don't he gets the last word. But then again, I am the one who is trying to push him places he doesn't want to go, so it shouldn't surprise me when whatever self-protection mechanisms are at play are employed.
Sorry for the stream of consciousness. I don't know exactly where I was trying to go when I started.