Step 1: Use the property of the tangent from the directrix side:
The origin lies on the directrix $x = 0$. Rather than use geometric properties of the directrix, work with slopes directly.
Step 2: Coordinates:
Parabola: $y^2 = 2a x - a^2$. Differentiate: $2yy' = 2a$, so slope at $P(\alpha,4)$ is $a/4$.
The tangent through the origin and $P$ has slope $4/\alpha$. Equate: $a/4 = 4/\alpha$, so $a\alpha = 16$.
Step 3: Combine with P on the curve:
$16 = 2a\alpha - a^2 = 32 - a^2$, so $a^2 = 16$, $a = 4$, and then $\alpha = 4$.
$OP = \sqrt{16+16} = \sqrt{32}$.
Final Answer:
$\sqrt{32}$, option (D).
\[ \boxed{\sqrt{32} \text{ (D)}} \]