Herbert Spencer somewhere says that the parabola of a ballistic object is actually a portion of an ellipse that is indistinguishable from a parabola--is that true? It would seem plausible since satellite orbits are ellipses and artillery trajectories are interrupted orbits.
The difference between the two cases is the direction of the gravity vector. If gravity is pulling towards a point (as we see in orbital mechanics), ballistic objects follow an elliptical (or sometimes hyperbolic) path. If, however, gravity points in a constant direction (as we often assume in terrestrial physics problems: it pulls "down"), we get a parabolic trajectory.
On the timescales of these trajectories that we call parabolic, the difference in direction of gravity from start to end of the flight is so tremendously minimal, that we can treat it as a perturbation from the "down" vector and then ignore it entirely. This works until the object is flying fast enough that the changing gravity vector starts to have a non-trivial effect.
At orbital velocities, the effect is so non-trivial that we don't even try to model it as a "down" vector plus a perturbation. We just model the vector pointing towards the center of the gravitational body.
One easy way to tell the difference between a highly eccentric elliptical orbit and a true parabolic orbit is that an object in a parabolic orbit travels at its escape velocity exactly. In astronomy, such orbits are as rare as circular orbits, i.e. they don't exist. An object well below the escape velocity can be in an elliptical orbit that has an eccentricity very close to 1, making it look much like a parabolic orbit when only part of the curve is examined.
A relatively slow projectile on the surface of the Earth is actually a closed curve ellipse, and if the Earth got out of its way by shrinking to the size of basketball with the same gravitational field, the object would return to its original place in a long cigar shaped elliptical path.
As an aside, if an object is traveling faster than its escape velocity, it is in a hyperbolic orbit.