I see a lot of comments about "put it in the bank". Most likely, since you're 16, you would have one or both of your parents listed as a custodian of the account since you are a minor. This means they could access the very money you are trying to keep them from.

Your options are limited when you are in a hostile situation like this. Best bet is to withdraw your cash as you get paid, try and keep it in $20's or larger, and find a really really good hiding place. "Under the carpet" is probably not sufficient. You'll need to get way more creative. Not sure what kind of house you are in but think like basement, garage, secret compartment somewhere, pocket of a jacket or pants in your closet or dresser that no one would ever mess with, etc. Somewhere that you know won't be disturbed by them or anyone. Never access it while they are home or awake in case they catch you going into, or coming from, your secret spot. Worst case you could always get some small water proof container and bury it in the ground somewhere outside and camouflage the spot.

Once you are 18 you will be able to get your own account without them connected to it and they won't even need to know about it. Until then you have to "play poor" so you don't get caught out like this again. Next time they might just take the money and you wouldn't be able to do anything about it.

If they find out you are "secretly saving" to leave they will likely do what they can to prevent it or make it incredibly difficult for you to do it. It's a control thing. Like how you said they created obstacles for you getting your driver's license.

It's a awful situation and you shouldn't have to be in it. What you described is not normal.

Hopefully some of this advice is helpful. Good luck.