It has arbitrary orientation in 2D.
I thought to equate the formulas for the arc lengths (s) between the midpoint and each end point from http://en.wikipedia.org/wiki/Parabola#Length_of_an_arc_of_a_parabola :
$$ h={p \over 2} $$
$$ q = \sqrt {f^2+h^2} $$
$$ s = {{hq}\over f} + f \space ln{({{h+q} \over f})} $$
where $p$ is the perpendicular distance from the point to the axis of symmetry and $f$ is the focal length. This leaves an equation with 3 unknowns - $p$ for each point and $f$. I hope to find other formulas involving those same variables to lock it down.
EDIT: I've realized that I don't need to solve this problem after all. It seems to have come from some sloppy documentation. It was supposed to be the midpoint in a special coordinate system used to describe it parametrically, which turns out to be quite easy.