How do I explain about non-degree options are worthwhile again?

Research the role you desire to apply for. Then research how to get there. What qualifications do you need? Do you need to have any side projects to show your experience? Do you have the knowledge to make them?
If not, what courses (there are free ones as well as those you pay) do you need to do? Are you able to complete it alongside your other responsibilities?
If your looking into apprenticeships, look onto the pros and cons. See how they can be applied to your daily life. First show to your mother that you have thought it out and have a plan of how your going to get to your ideal role. However, this is mainly fir yourself as well. Considering how you stated you have dropped your masters, you are at an age where you can make decisions yourself. Where do you want to be in a couple of years time, and how are you going to get there?
Whilst I understand the need to gain a parents approval and understanding, make sure you understand it first. This is because at the end of the day, you are the one living your life, and whilst it can be nice to follow what others plan for you, that isn't necessarily the right way.
On a last note, seeing as how you know how to get your name out there, do it. Prove to not only your parents but yourself that what your saying is correct. Show them that you don't necessarily need a degree to be successful.