The statement that describes the vertical shift when the parent function \( y = x \) is transformed to \( y = x + 5 \) is:
"The graph of the function \( y = x + 5 \) is the same as the graph of the parent function \( y = x \), but it is shifted vertically upward by 5 units."
In this transformation, every point on the original line \( y = x \) moves up by 5, resulting in a new line that maintains the same slope but is positioned higher on the Cartesian plane.