Personally I would talk to your parents, they support your queer sister and you have already come out to them as trans and they support you with your name change - you need the medical support, so talk to them. If that doesn't go well, then looking into DIY becomes more necessary - but don't start with DIY when you have a much better and safer alternative right in front of you.

It's scary, but many aspects of transition are scary, unfortunately.

One way to deal with the fear is to organize yourself, do lots of research, and then present your parents with an educated and reasonable argument. Preparation is one way to deal with anxiety and fear, basically.

To help, here is a literature review that summarizes the clinical outcomes of gender affirming care:

https://whatweknow.inequality.cornell.edu/topics/lgbt-equality/what-does-the-scholarly-research-say-about-the-well-being-of-transgender-people/

You can also point to the fact that every major medical association endorses gender affirming care for minors and adults:

https://glaad.org/medical-association-statements-supporting-trans-youth-healthcare-and-against-discriminatory/

And I also found this article by a bioethicist useful - it lays out some of the history of gender affirming care in the U.S. and debunks common misconceptions about gender affirming care and in a moderate tone - would be helpful for parents I would imagine:

https://www.openmindmag.org/articles/care-not-controversy

You could also try to find a trans-affirming therapist and see if they would be willing to moderate the conversation with your parents for you.