Do the distances from directrix and focus in a parabola shaped trajectory have a physical meaning?

The trajectory of a thrown object is secretly an ellipse, not a parabola, though the eccentricity of the "orbit" of a typical thrown object is so close to $e=1$ that it's hard to distinguish the two. At the other focus of the ellipse is Earth's center of mass; the focus of the parabola is the ellipse's empty focus.