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When you throw something on earth and calculate it as a parabola, that's based on the approximation that the force of gravity is constant across the trajectory. It's actually a tiny bit weaker 20 feet up than at ground level, but the difference is so small that you can ignore it. The trajectory in a constant gravational field is parabolic.
When something is high enough that you're doing orbital calculations, you have to take the inverse square law into account. The elliptical orbit is a consequence of the gravitational force being inversely proportional to the square of the distance. It doesn't apply if the force is constant.
You can also say, look at a small segment of the pointy part of a very skinny ellipse, and it will look approximately like a parabola.
Great explanation!