5. In a regular 2006-gon, the number of diagonals that are not parallel to any side is ( ).
(A) 2006
(B) $1003^{2}$
(C) $1003^{2}-1003$
(D) $1003^{2}-1002$
5.C.

For a regular $2n$-gon $A_{1} A_{2} \cdots A_{2 n}$, the number of diagonals is
$$
\frac{1}{2} \times 2 n(2 n-3)=n(2 n-3) \text {. }
$$

To calculate the number of diagonals parallel to side $A_{1} A_{2}$, since $A_{1} A_{2} \parallel A_{n+1} A_{n+2}$, the endpoints of the diagonals parallel to $A_{1} A_{2}$ can only be chosen from $2 n-4$ points, resulting in $n-2$ parallel lines. Therefore, the number of diagonals parallel to any side is $n(n-2)$. Consequently, the number of diagonals that are not parallel to any side is $n(2 n-3)-n(n-2)=n(n-1)$ lines.