I had a relative show me the ropes on the first day of the gym, so I had a bit of help in overcoming my initial gym-anxiety (and I waited like a 1½ decade more than you before I first went to the gym lol). But in hindsight what you're feeling is such a non-issue, and I know you will be feeling the same when you've gotten in the habit of going to the gym. Most people won't notice you're new, and noone will judge you for it. Seeing people start to work out makes me low key happy because it's one of the things that have always been there for me.

My tip for dealing with "gym anxiety" would to take it step by step. First time you're there, beeline for a threadmill/elliptical/etc. Do that for a while, meanwhile you're checking out the gym and trying to figure where everything is and getting comfortable with your surroundings. Then add exercises/machines one at a time. Small steps

And coerce your parents into providing a mode of transportation for you, getting into healthy habits at a young age is important as all hell as it'll have lasting effects for your entire life. Good luck!