Here is what I think may be quite helpful to challenge common misconceptions and reassure parents :
- show them you know what you're talking about : that's not something that just popped in your mind, but something you thought about thoroughly and seriously.
- Talk casually, without shame or anything like that : it's not a pathology / mental illness, it's not as uncommon as it seems, it doesn't make you monstruous or anything like that, trans people are like everybody else, and most of them you don't even notice they're trans (yeah that kinda suck to reassure them about cispassing, but that oftenly scare parents a lot, because they fear of social stigma)
- Challenge their misconceptions, and and if your family is made of quite rational and logical people, you can even give them some statistics about regrets (that seems to be the first fear of parents : "what if you regret this ?!") etc.
- no need to talk too much about details (like intimate things as genitals, or graphic description of surgeries you may consider, etc. You can reassure them about hormons, if you plan to take them, to describe simply what's gonna happen and most of all : that it will be progressive, slow and that it won't change who you are as a person.
- open yourself to them, show them this CO is a very biiiig sign of trust and explain how it's important for you to be supported by them, and how it's crucial for you to be respected as who you are. More chances to get people's empathy if you express your feelings.. It's very hard to come-out, and it's really a risk to lose the people we love.... so it can be very well taken if you show them that well, it's not easy for you, and you tell them because you need and trust them.
- It's important also NOT to ask for their opinion or permission. It's YOUR decision to transition (if you transition), and you don't need anyone's opinion about it. You tell them you're trans, you're not asking for the right to act on it. It's important because it shows they don't have a word in this. That mean they don't have any responsibility to take : they can only support you, or not.
- You can tell them what it means for you. They don't need to know how things are for "trans people", they're interested in YOU, so be specific about yourself, no others. You can for instance telle them that it means you are going to change your name, and that from now on you're gonna use this or that pronoun, and be gendered as ***, that you'd like them to put away the family photos in the living room for instance, or that you'd like them to stop calling you "son" or "my princess" etc. :)
- Finally.. lots of parents will think they did something wrong, or that they "should have known"... so it can be useful to say that :
1) there is nothing wrong with being trans, it just happens,
2) they didn't do anymore "wrong" than if you were left-handed instead of right-handed : that's just how you are, period.
3) People don't "become trans" because of education or things like that
4) it's ok if they didn't noticed it before because well, maybe you didn't yourself, or maybe you showed them what you wanted to show them, etc...
You can also give them ressources, like pamphlet about transidentity explained to family, or contact of organisation where they can be listened, meet other parents, forums on the internet etc.. It can be of GREAT help :)
That's just some ideas, you just do your best and what fits your situation the best. :)