[Dominick]

Orbits can be Parabolic, elliptical or hyperbolic. This was probably a elliptical, but it could have been parabolic. It is hard to tell from here. :-)

[Robert A Cook PE]

If you think about, a computed "orbit" (with any classic math shape (elliptical, hyperbolic, or parabola!) cannot have any "powered phase" or "controlled phase" - a calculated orbit is strictly "freefall" of an ideal mass moving about an ideal point mass at zero external gravity and no external forces...

Practically speaking: each curve works for small parts of each phase: boost phase with constantly reducing mass and air drag, boost phase with air drag and reducing mass, coasting (in vacuum towards the peak) at constant mass, coasting into atmosphere without spinning, coasting in high atmosphere with "only" spin drag, coasting in high atmosphere with controlled but stable flight, coasting under controlled "glider" flight.

[RightWhale]

Not sure about parabolic, but in a central gravitation field it is elliptical or hyperbolic. You can tell the difference by whether your spaceship comes back or not if it doesn't run into something.