Good point; this becomes more obvious if you imagine throwing the ball up and then immediately collapsing all the mass of the Earth into a single point at the centre. What path does the ball follow now? It's probably following a path we would more usually call an "orbit", and it sure looks a lot like an ellipse. Now just put the mass of the Earth back where it was, and notice that the ball hits the ground before it can trace out very much of its orbit.
Despite both being conic sections, cutting up an ellipse won't yield you parabolas. An ellipse has two focal points to which the sum of the distances is constant, while a parabola has a focal point and a directrix line to which the difference of the distances is constantly 0. Two different things.
More symmetrically, Every comic section has one focus and one directrix (and a semi-major axis for scale), eccentricity is the ratio of distance. Ellipse has eccentricity less than 1, and a parabola has eccentricity equal to 1.
