My parents were actually the ones who first suspected it when I was about 16 or 17, so they never had issues with coming to grips with it; if anything, I was the one who had problems accepting it. I remember the first time my parents sat me down and explained what Aspergers was (my first reaction was laughing at the name, "ass-burgers", which my parents thought was a bizarre reaction but how else was an immature 16-17 y/o going to react to that?), and finding it downright insulting that they would think of me as "socially crippled" (as I thought of it back then). Even though it was staring me right in the face, I swept the issue under the rug and carried on as if I was a perfectly normal, socially functioning human being (which I was anything but). Took me a good 4 or 5 years when it got to the point that I couldn't ignore it anymore (the inability to make close friends, communication issues with my parents, etc) that I finally had the maturity to come to terms with it.
I feel very fortunate in that my parents did me a huge favor in the end by bringing it to my attention (which allowed me to become a better self-observer and hone in on areas that needed improvement), but it's sad how much of a disservice some parents do to their kids by approaching it with denial or flat-out hostility.
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"All I can do is be me, whoever that is." - Bob Dylan