Explain that for the parabola $y^{2}=a x(a \neq 0)$, the coordinates of the focus $F$ are $\left(\frac{a}{4}, 0\right)$. Therefore, the equation of the line $l$ is $y=2\left(x-\frac{a}{4}\right)$, and its intersection with the $y$-axis is $A\left(0,-\frac{a}{2}\right)$. Thus, the area of $\triangle O A F$ is $\frac{1}{2}$ $\left|\frac{a}{4}\right| \cdot\left|\frac{a}{2}\right|=4$, solving for $a$ gives $a= \pm 8$. Hence, the equation of the parabola is $y^{2}=$ $\pm 8 x$. Choose B.