Yes, putting your name and Social Security number in the Dependents section of your parents' tax return means that they claimed you as a dependent. It doesn't matter whether any of the boxes are checked.
If you are not a dependent of your parents for 2019, then yes, they have to file an amended return.
You will have to file your tax return by mail. It will be processed normally. You don't have to wait for your parents' amended return to be processed. If they file the amended return reasonably soon, the IRS will figure it out and you won't have any problems.
