My parents don't accept this either and tbh, for a lot of the same reasons.
But here's what I've found to be reality:
1) People viewed me as odd before I started any of the transition stage. While there is nothing wrong with being a butch woman, it does gets attention and for me that attention wasn't justifying in my masculinity, it just made me feel like I was wrong because I didn't feel like someone breaking the norms or attacking binary, I was a man that stood out for being something I didn't want to be. Since transitioning I actually finally found what I needed in life: normality! I went from being that "weirdo girl" to just being plain old, every day, Jack! For all that both our parental units fear, I find the absolute opposite happened. I went from being perceived as odd to just being normal!
2) Employers don't even have to know once everything is changed really. And when I came out at work (I was returning to a place I had previously worked at with a female name) they were entirely supportive. So supportive in fact that even though I'm working with people I know, I go to work stealth. I kid you not! Unless people knew me from before, they have no idea I am a transguy and no one tells them! Not ONE person has told anyone I work with. I keep waiting for it but every interaction I can imagine happening has already happened, even when alcohol is involved (Xmas party) and no one has slipped up or dropped hints. I'm prepared for questions if it happens but having this time at least to establish a baseline of, "Well up until now you had no idea, so really, what difference does it make?" I find useful.
3) I teach a TKD class where a fair amount of the parents know I am trans and they let me teach their impressionable kids. My instructor didn't care one jot when I came out, and no parents pulled their kids when they heard I would be assisting teaching or teaching their kids. And my instructor didn't tell anyone who didn't know because it's none of their concern!
4) People can suffer selective blindness/deafness. Seriously, a few weeks before I started T a new guy joined our Taekwon-do class. 7 months on we're pretty much best friends and the guy has no idea I'm a transguy or at the very least cares so little he doesn't say anything and tbh, just based on some of the convos we've had, I'm leaning heavily on the former. He met me pre T, before my voice dropped, and he thinks I'm just a regular guy! Ha!
5) It's easy to lie or embellish truth to protect yourself in extreme circumstances. People tend to have empathy for intersex individuals over trans individuals. As far as I see it, all trans individuals are intersex really because there's obviously a conflict in that area and it's just an argument of the mind over physical makeup. And heck, not everyone has been tested for intersex conditions so who knows?! If people find out and start on the "freak" stuff, you can just say you were born intersex and had chemical processes induced that turned out to be the wrong way, would they treat someone born paralysed the same way? None of that is a lie, regardless of situation. Testosterone and oestrogen forces the chemical process that makes us develop in defining male of female ways. But, put it that way and they think it means you had surgery or treatments that made you into what turned out to be the wrong way and at that point many will back off just out of fear of offending. Now, this is controversial because some people will hate it as it makes trans individuals seem less justified in their transition. I view it more as looking out for number 1, and that it's not even lying really by keeping it so vague. Just like if I'm asked about chest scars in the future: "I had non cancerous benign growths removed, no biggy!" Medically that's actually true!
People aren't all out there with pitch forks ready to get us. The fear is often unjustified and more something that should be feared with lack of treatment IMO!
The good thing is that your mum is coming from a stance of concern for you as a result of this. She fears others' reaction to you, it seems, more than how she actually feels, although that may be hard to see right now. And that's a good thing because it means she still cares for you and that's great! The problem is the generation gap. Even if she's relatively young as a parent, things have really changed these days! And quite drastically so. People of her generation are themselves more open, and as generations continue, acceptance improves. It's her elders that will be more likely to be non-accepting and that's influencing her concerns over what your own peers (or even hers) are more likely to think.
The advantage for us is there's at least the option to remove that which makes us stand out if we choose to do so (going stealth). If you want T and top surgery, those are processes that help with passing. They help you stand out less if that is your wish and if that is something you wish for yourself, the reality is that it will make everything much easier for you in terms of acceptance and employ-ability as well as avoiding discrimination. I mean, who will know unless you tell them? It may help if you show them some examples of transguys as we can pass damn well! I don't wish to cause offence with the following but the go to image most people have when they hear of trans individuals is to imaging MTFs first, or at least what they perceive as MTFs which tends to be more an example of drag queens. And extreme cases, they start imagining the most masculine, buff looking guy they can think of in a dress. They have Arnold Schwarzenegger in a dress like something out of the movie 'Junior' in their head at this point. But that's not how it goes and we know that. They see drag queen before trans individuals and see flamboyancy and fear for you as a result of that unrealistic image, even if they know who you are as a person. The reality is you have a very good chance of passing well enough that no one questions your gender at all. Now, trying to get this across will be difficult but I found what my parents worried about was more what I faced BEFORE T as opposed to how much easier it is today.
At the end of the day, sometimes the best way is to do what is right for you and prove away their fears when they see the reality of the situation.
