Answer:
A freely falling body, as the name implies, is not hindered in its fall. "Not hindered" is to be understood as not appreciatively hindered for the purposes of describing its motion with a simple equation. A relatively heavy object near the Earth is not hinder for a short trajectory of a few meters. Then, a simple rock or ball or anything, even a person, will move in a straight line or in an arc that is well approximated by a parabola. (The actual path of a freely moving object will be an ellipse, but the short portion you see in a trajectory near Earth is indistinguishable from a parabola.) If you want a purer form of the freely falling object, the best examples are bodies outside the Earth's atmosphere, for example, satellites that go around the Earth. These circular orbits are simplified versions of an ellipse.
For extra credit, explain how a geostationary satellite, which appears to remain at the same point in the sky above the equator, is actually moving in an ellipse.