I've been out to my parents for several years now and I came out to them a bit later in my life (mid-30s). I've been very lucky with having supportive parental units, but I though I'd share a few of the -funnier? unfortunate? - things they said early on.
My Mom
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I strategically told my mother first, as I knew she'd be the more receptive of the two.
About an hour after telling here and talking about it with her:
Mom: "You know, I always knew something wasn't quite right about you. [pause] That didn't sound right, but you know what I mean. You've always marched to the beat of a different drummer."
She also said at a later point that same weekend:
Mom: "I'm glad you told me. This is going to be selfish, but I feel like this barrier between us is gone now that you've told me. It's like there was always something you weren't saying or sharing, always keeping everyone at an arms length. I feel like that's gone now. I feel like we can move on now."
Later that year when I was visiting them:
Mom: "You're happy now. You were always so unhappy, as a child, as an adult. You smile now, you talk to people now. It's like you're a different person. I'm really glad that's how it is now, but I sometimes feel like I've lost my daughter and it makes me really sad."
(I felt bad after she said that, pre-T me would have bawled like a baby (post-T me has a difficult time crying at all). I didn't want her to feel sad, you know? I knew that she needed to, for lack of a better way to put it, mourn, but I spent a few weeks after that trying to figure out if there was a way I could make it any easier for her.)
Earlier this year:
Mom: "You know, when you are not here, I sometimes slip up and call you by your old name or she. Your father is worse, especially if we're talking about something that happened in the past. But when you are here in front of me, well, ...there's just nothing female about you. It would be weird to think of someone that sounds like you or acts like you or looks like you as
she."
Thanks for the loving support mom; albeit a little too black-and-white with male and female behaviors/roles still, she's getting there and has been very supportive. Just says some great, slightly awkward things sometimes. I educate when appropriate.

My Dad
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My father has had a much harder time with it, but has overall been more supportive than I would have expected. My mother told him after the above conversation, and the first conversation we had after that included:
Dad: I'm always going to care about you, I just want to make sure you aren't really screwing up your life with this.
Later in the conversation:
Dad: I don't know, I don't think it's (transgender/GID) real.
Me: What do you think it is then?
Dad: [pause] I think it's a last resort, when people have failed at fitting well into society or in relationships as a man or a woman.
Ouch, thanks dad.
I did then ask him if that was his polite and slightly indirect way of telling me he thinks I failed at fitting in and finding a relationship and he immediately back-pedaled (as I have a good career, a good social circle, and a loving girlfriend). He then said he needed to think about it more and I told him I would send him some reference reading to help out. (I did.)
He's gotten better about it. He's managed to use my chosen name pretty reliably the last several times I've talked to him. He slipped up and called me his daughter not too long ago and immediately corrected himself awkwardly with son. It's the little things.
I'll have to post in the "Stuff Stranger's Say" thread, I've got some unfortunate/funny ones for that category.