True or false if line x is parallel to line y, then point z is the only point through which y passes

The context of the question is not very clear.

If Z is the only point available, then yes, this the the only point line y passes.

But since y is long, we could add points A, B, ..... (see attached picture).

So the answer is very much context dependent.

On the other hand, if the question had been, if line y is parallel to line x, then line y is the only LINE that passes through point Z, then the answer is true.