How did you end up with r = (x, sin(y), z)? If you take x and z to be the parameters, wouldn't y = cos-1√(x^2+z^2) ?
When parameterizing a surface, you use two variables, right?
With what I did, my surface ended up being parametrized as \vec{r}(x,y) = <x, y, \sqrt{cos^2y-x^2}>