3. 2 .

Let $A\left(s^{2}+1,2 s\right), B\left(t, t^{2}+4\right)$. Then $l_{O B}:\left(t^{2}+4\right) x-t y=0$.
Let the distance from point $A$ to line $O B$ be $h$, then
$$
\begin{array}{l}
h=\frac{\left|\left(t^{2}+4\right)\left(s^{2}+1\right)-t \cdot 2 s\right|}{\sqrt{\left(t^{2}+4\right)^{2}+t^{2}}} . \\
\text { Hence } S_{\triangle O A B}=\frac{1}{2} O B h \\
=\frac{1}{2}\left|\left(t^{2}+4\right)\left(s^{2}+1\right)-2 s t\right| \\
=\frac{1}{2}\left[\left(s^{2} t^{2}+3 s^{2}+(s-t)^{2}+4\right] \geqslant 2 .\right.
\end{array}
$$

When $s=t=0$, the equality holds.
3. Let $O$ be the origin, $A$ be a moving point on the parabola $x=\frac{1}{4} y^{2}+1$, and $B$ be a moving point on the parabola $y=x^{2}+4$. Then the minimum value of the area of $\triangle O A B$ is $\qquad$ .