The planet's curvature is not very relevant to the article, but, yes, if an object is in a free fall at a slow speed (I mean non-orbital) its trajectory is:
- parabola if you assume flat&infinite ground,
- ellipsis if you assume a spherical planet (an ellipsis is crossing the planet's surface).
Of course, a very very eccentric ellipse approximates a parabola quite well. You only start to notice errors when dealing with trajectories dozens of miles long (on Earth).
