I'm not sure if it is because I was an male only child raised by my mother and grandmother with no good male role models in my life... but I've always disliked the whole "your genitalia defines who you are" commonplace perspective that permeates throughout our society.
I'm not saying that this should be ignored or denied. What I mean is that it is only an aspect of who you are as a person and as an individual. It is not THE most critical or dominant part of WHO you are as a personality. Just merely one part of the great many other things that all combined together make you who "you" are.
Anyway, what you are describing sounds so difficult. I cannot imagine what that would feel like.
I imagine that your parents would also struggle to understand what it would feel like also. It is difficult to grasp concepts that you have no personal point of reference to be able to comprehend or process them. With time, perseverance and the desire to it is possible for people to at least basically comprehend... but it is a long road for most to get there. Perhaps this is why it seems that your parents don't think that it is "a big deal"... it might matter to them a great deal because they love you, but they are still trying to wrap their heads around where you are coming from and just haven't quite managed to get there yet.
Maybe if you described it to them from a different angle or in different ways it might "click" for them and they will understand better where you are coming from and why this is a big issue for you. From your brief description it does sound like you are fortunate enough to have parents that want to understand... many people are not so lucky.
I have been told that I also sometimes come across as cold or emotionless. It's more that my brain is always in a state of "processing" lots of information that I am inside my own head a lot even while I'm in the company of others... and outwardly it might seem that I am "distant" or not fully engaged. Which is rarely true... I'm just mentally doing multiple things at the same time. Perhaps this is the same for you.
I've never "liked" being male. Not as a kid, not as an adult. Males seem to me like the crappier gender of the two. I often have felt disgusted by my own gender and that I am a "member" of it. I don't feel any pride in "masculinity" whatsoever. Never have. Not the same as what you are describing of course but I do understand feeling "disgust" over certain aspects of yourself.
I hope things work out.