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.
Lots of answers touched the correct answer, which is that in reality things don’t follow parabolas on earth, a parabola is just close enough to the actual thing the object is doing to be indistinguishable. In reality everything follows elliptical orbits, but the top of an ellipsis with a Major axis of 6378 km and a few meters in the minor axis looks the same as a parabola, especially when you don’t see the full orbit because the object hits the ground. If you were to throw a rock and suddenly the entire earth besides that rock collapsed to a single point, your rock will orbit earth in an elliptical orbit.