1. **Problem statement:** Given points $A(3,1,2)$, $B(0,1,-2)$, and $C(1,2,0)$, find point $D$ such that $ABCD$ forms a parallelogram. 2. **Formula and concept:** In vector geometry, for $ABCD$ to be a parallelogram, the vector $\overrightarrow{AD}$ must equal the vector $\overrightarrow{BC}$. This means: $$\overrightarrow{D} = \overrightarrow{A} + \overrightarrow{BC}$$ where $$\overrightarrow{BC} = \overrightarrow{C} - \overrightarrow{B}$$ 3. **Calculate vector $\overrightarrow{BC}$:** $$\overrightarrow{BC} = (1 - 0, 2 - 1, 0 - (-2)) = (1, 1, 2)$$ 4. **Find coordinates of $D$:** $$D = A + BC = (3,1,2) + (1,1,2) = (3+1, 1+1, 2+2) = (4, 2, 4)$$ 5. **Answer:** The coordinates of point $D$ are $\boxed{(4, 2, 4)}$. This ensures $ABCD$ is a parallelogram because opposite sides are equal vectors.