You cannot make your parents understand being non-binary in one conversation, and you do not have to. Tell them the truth simply, give them time to catch up, and judge them by where they land in a year, not by their first stunned reaction.
I want to tell you about someone I think of as Jay.
Jay is nineteen and came home from their first year at university with a haircut, a nose stud, and a sentence they had been practising on the train for three hours. Over dinner, somewhere between the potatoes and the point where it would have been too late to say anything at all, they said it. "I am non-binary. I am not your daughter and I am not your son. I use they and them now."
There was a pause of the particular kind that families specialise in. Then Jay's dad said, "But you are our girl," not unkindly, more bewildered, as though Jay had announced they were moving to the moon. And Jay's mum said nothing at all, and got up to clear plates that were not finished, which is its own language.
If Jay were sitting with me a few days later, deflated, they would say, "I did it wrong. I made it weird. They did not get it."
So let me say to you what I would say to Jay. You cannot make someone understand non-binary in a single dinner, and it is not a failure that they did not. You have lived with this truth for years. Your parents heard it for the first time roughly between two mouthfuls. They are not behind because they are bad. They are behind because they just started, and you are asking them to catch up to a place it took you a long time to reach. Give them the time you gave yourself.
Non-binary people are not new, and they are not a fashion that arrived with your generation. People have lived outside the two boxes in every culture and every century we have records for. What is new is only the freedom to say it at nineteen, over dinner, in your parents' house. Your parents did not grow up with that freedom or that language, so the words feel strange in their mouths. Strange is not the same as unwilling.
What does a parent actually have to do? Less than they fear. Use the name. Try the pronouns, get them wrong, correct yourself, carry on. Nobody gets they and them right immediately, and the trans and non-binary people I know are far more patient with an honest fumble than with a stony refusal. The slip followed by a quiet "sorry, them" is the sound of someone trying, and trying is most of it.
I told Jay to expect grief, too, and not to take it as rejection. A parent sometimes has to let go of a daughter or a son they had imagined, a wedding they had pictured, a version of the future that lived in their head. That grief is about their picture, not about you being wrong. Let them have it without letting it become your problem to fix. The difficulty in the room is the adjusting, not the identity.
It was Jay's mum who came round first, which is often the way. She started leaving the right name on the notes she stuck to the fridge. She caught herself mid-word on the phone, "she, sorry, they". Small things. They are never small to the person they are aimed at.
Jay's dad took longer, and there were a couple of conversations that went badly before any of them went well. I will not tell you every family lands softly, because some do not, and if yours is one of the harder ones then you build your family from the people who do see you, and you do not wait alone in the meantime. But ditch nobody too soon. People surprise you, usually slowest of all the ones who love you most.
On Jay's twentieth birthday, the card from home was signed, in their dad's blocky handwriting, "To my kid Jay, love Dad." Not son, not daughter. Kid. He had reached for the truest word he had, and found it, and it was clumsy and it was perfect, and Jay kept the card.