The only way to do it is just do it. Don't beat around the bush and be upfront and direct, and remember you can only make it more awkward if you do stumble around and try to let people connect the dots.
And sometimes people will surprise you. I was so worried my own parents would be very negative and harsh about it, also based on things they have said in the past, but it turns out they are both supportive of me (my dad even said he doesn't like it but he will be there for me).
Mentally, it was hard to do. I did use a particular event to bring it up though. I developed IBS about a year ago, and I had a colonoscopy a couple of months ago that revealed a couple of pre-cancerous polyps, one that was big enough to be in the "high risk" area. So having come that close to cancer, I let my parents know the results, and that it got me thinking I have to move on with my life and do this. They are still absorbing the information, and my mom keeps asking me if it's some phase, or if I'm just a cross dresser, or if sure because she has never perceived me as being feminine, but it went much better than I could have ever planned on or hoped for, and not only did it get me out of the closet to them, it left me feeling that I could finally bury the past and move beyond my mom having been emotionally abusive and my dad walking out one night, which was a very nice extra that I didn't anticipate.