C

To derive the trajectory of a projectile fired horizontally, we consider the equations of motion under constant acceleration due to gravity. The horizontal motion does not affect the vertical motion, which is given by the equation:
$$y = h - \frac{1}{2} g t^2$$
where h is the initial height and g is the acceleration due to gravity. This gives us a parabolic equation, confirming that the trajectory is indeed a parabola. Therefore, the correct answer is Option C.