To find the coordinates of vertex D' after dilating the parallelogram with respect to the origin using a magnitude of \( \frac{1}{2} \), we multiply each coordinate of vertex D by \( \frac{1}{2} \).
Vertex D has coordinates (2, -2).
Applying the dilation:
\[ D' = \left( \frac{1}{2} \times 2, \frac{1}{2} \times -2 \right) = (1, -1) \]
Therefore, the coordinates of vertex D' after the dilation are \( (1, -1) \).