I don't think you've quite clocked it. It's not that one of the statements has to be wrong, because that's just a point in the cycle. If A is wrong, then B is right, which means A is right, which means B is wrong, which means A is wrong and the cycle begins anew.
They aren't wrong, they're contradictory. There is no logical way to parse the two statements together. That's what a paradox is.
Neither statement can be true OR false. If statement A is true, statement B is true, which means statement A is false. To simplify, if statement A is true, statement A is false.
"This statement is false" can be neither true nor false. That is the most basic paradox there is.