Hi there Elizabeth, welcome!
The fact that your parents think Trans people are "confused" rather than a sin against God or some such thing is a good sign. My parents felt the same way about trans people. When I came out to them, they didn't react with anger or any of the emotions that you would have thought a somewhat conservative family would have reacted. My parents always stressed that they would love me know matter "what" I was, which turned out to be true. I think this goes for a lot of parents - they are always outspoken against it until someone they know and care about comes out as trans. I think if you take the time to explain it to them, and have them read up on it, they will be more likely to understand where you're coming from.