It's possible that crying in an argumentative situation with your parents is more of a learned reaction, not so much a deliberate manipulation. You learned it by actually crying for real when you were quite little, and then when you were a bit bigger but still young, got rewarded for it (by staying home from school). That seems like it would cement the tears-and-sadness reaction even though it's not logical or very useful now. I'm thinking of when a little child is afraid of something (bees, for example) for a reason (such as, once was stung) and still as an adult has a knee-jerk fear reaction to them even if he knows intellectually that bees aren't interested in stinging them. Often the way through that (and to get the knee-jerk reaction to go away) is by getting information. (Reading a book on the life of bees, for example, has cured people of phobias about getting stung.)
It might be interesting to look up "learned fear responses" or "learned emotional reactions," to see if anything is being written about how to break your reactive pattern. One way to get past being overly reactive is to simply act differently next time and see what happens. In other words, react with some totally different way of talking or ignoring, or whatever. It can help to do creative visualization and/or imagine yourself telling them that you can't see any reason to waste your emotional energy on worry over this argument, or something like that.
You might also look up different problem-solving techniques, to try next time your dad gets under your skin. (It sounds like he hooks you more than your mom does.) I don't recommend giving any other person your emotional power, especially if it's just out of old, old habit.
The problem is, you're asking a pretty complicated psychological question and frankly, who knows? It could be learned behavior, but I'd think if it was why would you now only use it on your parents? Of course, that's possible, as psychological aspects of our lives are pretty weird and very hard to figure out cause and effect. So while it's interesting to figure out the why of things, given the rest of your life seems to be going pretty well and you don't seem to have some deep rooted anxiety or depression or other dysfunction that is adversely affecting your life every day I'd be more inclined to just use what you already know, that you're doing this and you don't want to do this, and try to stop doing it. One way would be to tell your folks that every time you start crying during a discussion, have them point out that, hey, you're doing it again. At your age, it's pretty easy to change things -- you're a lot older than you were when you were but you're really young still and very adaptable. Now, if you were still doing this in your thirties, that would be much harder of a habit to break. So yeah, you can spend a lot of time trying to figure it out, and you would benefit from doing so, but in the meantime, the cognitive behavioral approach is to try to fix it faster than that by learning practical mechanisms to stop doing it. Here's an example from my life that isn't completely related, but I sucked my thumb for a long time because my parents didn't stop me from doing it. A doctor just put a bandaid on it and that was that, I stopped quickly. Now, I could have tried to find out why I sucked my thumb for so long, but the bandaid was a lot quicker.