Let $a$ and $b$ be positive real numbers. In the Cartesian coordinate plane, using the parabola
$$p ( x ) = ( x - a ) ^ { 2 } - b$$
that passes through the origin, three parabolas defined as
$$\begin{aligned}
& p ( x + a ) + b \\
& p ( x + a ) - b \\
& p ( x - a ) - b
\end{aligned}$$
have their vertices at the vertices of a triangle with an area of 16 square units.
Accordingly, what is the sum $a + b$?
A) 6
B) 9
C) 12
D) 15
E) 18