So of course both of these are slight simplifications, but what is the connection between the two? If the earth is basically a circle, is an ellipse just a parabola stretched around a circle? Is a parabola just an approximation of a tiny part of an ellipse? How high do you have to be before you change your calculations of a trajectory?
The Math ain’t mathing.
Elliptical orbit is what you get when all gravity vectors point to a single source such as Earth’s center of mass. If you instead make the “flat Earth” assumption that all gravity vectors are parallel, then you get a parabola. Since most of us aren’t throwing things very far on a planetary scale, the parabolic approximation is frequently considered “good enough”.