The line $x - 2y + 1 = 0$ intersects the parabola $y^{2} = 2px \ (p > 0)$ at points $A , B$ with $AB = 4\sqrt{15}$ .
(1) Find the value of $p$ ;
(2) Let $F$ be the focus of $y^{2} = 2px$ . Let $M , N$ be two points on the parabola such that $\overrightarrow{MF} \perp \overrightarrow{NF}$ . Find the minimum area of $\triangle MNF$ .