They are lying to you.
When you are 18, you can have YOUR OWN checking account with YOUR OWN debit card that will let you use YOUR MONEY for ANYTHING that you want. At 18, you DO NOT need to have a joint account with anyone else. YOU can go to any bank and talk to a banker to set up your accounts WITHOUT anyone else with you. Your parents would never be on the account and if you are careful, they will never even know you have it. (have your statements set up as e-mailed instead of regular mail) The only way the account would be discovered is if your parents get the mail on the day your debit card arrives in the mail.
As for CREDIT card accounts, you need to apply for those and you need to get approved. In those cases, you MIGHT need to have a joint account with someone else BUT not always. If you have had a job for a couple years, you might be able to get approved for a card without anyone else on it. You also could save money and do a secured credit card. Those are easier to be approved for because you make a deposit into a savings account that the bank holds as collateral to let you have the credit card. For credit card accounts, you can also have the statements e-mailed. The only way your parents would know you have it is if they check the mail the day the card arrives in the mail.
At 18, YOU have choices and YOU have some control of your life. HOWEVER, remember one thing. As long as you live in their house, they will continue to try to control you. Legally, at 18, you can make your own choices. Many parents have a "my house, my rules" attitude so even if you get your own checking account, debit card, and credit card, they might still try to control what you are doing with them. But----their names aren't on it ----- so it will make it harder for them. If they take your cards - you report the cards as "lost" or "stolen" to the bank that issued them. (be sure to have the phone numbers from the backs of the cards saved in case this happens and have your card numbers wrote down someplace very safe). When you report a card as lost, the card is deactivated and a new one sent to you.
By the way - you can also avoid mail box issues by going to your local post office and getting a PO box. You can have your personal mail sent to the PO box and your parents would never see it.