Female Chest Dysphoria?

So here's my situation. I'm out as gay/lesbian because I know 100% that I'm attracted to women alone. It's great that I've finally come to terms with this. My only problem is that I don't really feel 100% comfortable as a female?

Now, I know I'm not transgender. I've always been a major tomboy, but I have no desire to actually be a guy, if that makes any sense. However, there's certain things I hate about my body, like my chest for example. I've never really minded being misgendered either because I consider myself androgynous. That's a fact that I've never told anyone because I don't feel like my gender expression needs to be shared with anyone.

I still live under my parents' roof, and I feel like they'd freak out if I told them that I wanted a binder. Hell, if I told them about any of this gender questioning stuff they'd still freak out. They're way more supportive with my sexuality than I expected them to be, but I can tell they don't really approve of being transgender. Weird, huh?

My main question here is, would it be strange for me to bind/get top surgery someday? I've hated my chest for as long as I can remember, and it's only getting worse as I grow older and find myself more. Any advice?

Also worth noting: I really like they/them pronouns, more than he/him. Any ideas as to what this all could be? I kinda feel like I'm still holding back the real me because of my transphobic parents.

(Sorry if this is a lot).