13. Let $A$ and $B$ be two distinct points on the parabola
$$
y^{2}=2 p x(p>0)
$$

Then the minimum value of $|\overrightarrow{O A}+\overrightarrow{O B}|^{2}-|\overrightarrow{A B}|^{2}$ is $\qquad$.
13. $-4 p^{2}$.

Let $A\left(x_{A}, y_{A}\right), B\left(x_{B}, y_{B}\right)$. Then
$$
\begin{array}{l}
|\overrightarrow{O A}+\overrightarrow{O B}|^{2}=\left(x_{A}+x_{B}\right)^{2}+\left(y_{A}+y_{B}\right)^{2}, \\
|\overrightarrow{A B}|^{2}=\left(x_{A}-x_{B}\right)^{2}+\left(y_{A}-y_{B}\right)^{2}, \\
|\overrightarrow{O A}+\overrightarrow{O B}|^{2}-|\overrightarrow{A B}|^{2}=4\left(x_{A} x_{B}+y_{A} y_{B}\right) .
\end{array}
$$

If the slope of line $A B$ exists (let it be $k$), then
$$
l_{A B}: y=k(x-a) \text {. }
$$

Substituting it into the parabola equation yields
$$
k^{2} x^{2}-2\left(a k^{2}+p\right) x+k^{2} a^{2}=0 \text {. }
$$

Then $x_{A} x_{B}=a^{2}$,
$$
y_{A} y_{B}=k^{2}\left(x_{A}-a\right)\left(x_{B}-a\right)=-2 a p \text {. }
$$

Thus, $|\overrightarrow{O A}+\overrightarrow{O B}|^{2}-|\overrightarrow{A B}|^{2}$
$$
=4\left[(a-p)^{2}-p^{2}\right] \geqslant-4 p^{2} \text {. }
$$

When the slope of line $A B$ does not exist, we have
$$
x_{A}=x_{B}=a, y_{A}=-y_{B}=\sqrt{2 a p} \text {, }
$$

Equation (1) also holds.
Equality in Equation (1) holds if and only if $a=p$.