So there's a lot of information I could try and pass on to you in response to some of those questions, but let me start by referring you here:
I would suggest taking a look around some of the research posts and reading then yourself. You can get a lot of information out of them!
There is also a great expelaination of everything modern science knows about trans individuals, broken down into something most people can understand here:
(Spoiler alert: trans people are scientifically valid.)
To answer one of your questions directly though, I tend to like to describe dysphoria to people as phantom limb syndrome. Your brain expects signals from X, or it actually gets signals from Y, and sometimes doesn't get signals at all. Your brain is smart. It knows inherently that there is a mismatch between what it's set up to be and what it's actully experiencing, and that's why when you look in the mirror it gets that much worse, or why it tends to be more obvious after puberty. Just like it with phantom limb syndrome, it provides visual confirmation that something is wrong and your brains responds by sending out distress signals. You don't have to know this on a conscious level, because your brain alerts you to this automatically, on a lower level, just like breathing. It's the fact that this is a lower level brain structure thing that makes it so difficult to describe.
Hopefully that makes sense? Best of luck with your parents.