I suppose a rectangular cuboid with edges parallel to the $x,y,z$ axis is what you mean by 'right-angled parallelpied'. For any rectangle $ABCD$ with edges parallel to the axis, there exists points $E,F,G,H,I,J,K,L$ such that $ABCD-EFGH$, $EFGH-IJKL$, $IJKL-ABCD$ are all right-angeld parallelpied. Then the number of red vertices among $ABCD = \frac{1}{2} ( (ABCDEFGH) - (EFGHIJKL) + (IJKLABCD) )$ is a multiple of $2$. Hence if the colors of $A,B,C$ are given, then we can determine the color of $D$. Now suppose that the color is given for all points on at least one of the $x,y,z$ axis. By the observation we have made, we can uniqely determine the colors for every point on the $xy$ plane, and then for every point in $E$. And one can check that the coloring made by this process satisfies the original condition. Therefore there are $2^{3 \cdot 1983 - 2}$ colorings.