1. Find the parabola parameter $a$ using point $P(4,4\sqrt{3})$

Given parabola: $y^2=4ax$ Since $P(4,4\sqrt{3})$ lies on it, $(4\sqrt{3})^2=4a(4)$ $48=16a$ $a=3$

So the parabola is $y^2=12x$ with focus $(a,0)=(3,0)$ and directrix $x=-3$

2. Parametric form of point $P$

For parabola $y^2=4ax$, a point is $(at^2,2at)$ Here $a=3$, so $(3t^2,6t)=(4,4\sqrt{3})$ From $6t=4\sqrt{3}$, $t=\frac{2\sqrt{3}}{3}$ Check: $3t^2=3\cdot \frac{4}{3}=4$ correct.

So parameter of $P$ is $t_1=\frac{2\sqrt{3}}{3}$

3. Find the other end $Q$ of the focal chord

For parabola $y^2=4ax$, if points with parameters $t_1,t_2$ form a focal chord, then $t_1t_2=-1$ Hence $t_2=-\frac{1}{t_1}=-\frac{\sqrt{3}}{2}$

Therefore, $Q=(3t_2^2,6t_2)=\left(3\cdot \frac{3}{4},6\cdot -\frac{\sqrt{3}}{2}\right)=\left(\frac{9}{4},-3\sqrt{3}\right)$

4. Find points $M$ and $N$ on the directrix

The directrix is the vertical line $x=-3$ So the foot of perpendicular from a point $(x,y)$ to the directrix is $(-3,y)$.

Thus, $M=(-3,4\sqrt{3})$ $N=(-3,-3\sqrt{3})$

So quadrilateral vertices are $P(4,4\sqrt{3}),\quad Q\left(\frac{9}{4},-3\sqrt{3}\right),\quad N(-3,-3\sqrt{3}),\quad M(-3,4\sqrt{3})$

5. Compute area of quadrilateral $PQMN$

We use the shoelace formula on vertices in order: $P(4,4\sqrt{3}),\ Q\left(\frac{9}{4},-3\sqrt{3}\right),\ N(-3,-3\sqrt{3}),\ M(-3,4\sqrt{3})$

Compute $S_1=4(-3\sqrt{3})+\frac{9}{4}(-3\sqrt{3})+(-3)(4\sqrt{3})+(-3)(4\sqrt{3})$ $=-12\sqrt{3}-\frac{27\sqrt{3}}{4}-12\sqrt{3}-12\sqrt{3}$ $=-\frac{171\sqrt{3}}{4}$

Compute $S_2=(4\sqrt{3})\left(\frac{9}{4}\right)+(-3\sqrt{3})(-3)+(-3\sqrt{3})(-3)+(4\sqrt{3})(4)$ $=9\sqrt{3}+9\sqrt{3}+9\sqrt{3}+16\sqrt{3}$ $=43\sqrt{3}$

Area is $\frac12|S_1-S_2|=\frac12\left| -\frac{171\sqrt{3}}{4}-43\sqrt{3} \right|$ $=\frac12\left| -\frac{171\sqrt{3}+172\sqrt{3}}{4} \right|$ $=\frac12\cdot \frac{343\sqrt{3}}{4}$ $=\frac{343\sqrt{3}}{8}$

6. Match with options

$\boxed{\frac{343\sqrt{3}}{8}}$ So the correct option is $\boxed{\text{B}}$

7. Comparison with stored answer

Stored correct answer: B

My derived answer is also B.