Are you familiar with the difference between polar (r, theta) coordinates and cartesian (x,y) coordinates? Parabolas are the solution to gravity that is uniform no matter where you sample. It assumes that gravity points in the same direction with the same magnitude no matter where you are. In this model, gravitational acceleration is always 9.81 m/s^2 in the -y direction. That is a reasonable simplification for things that are at the human scale constrained to near the earth’s surface. Deviations from air resistance will matter far more than errors stemming from that assumption anyway.

Conic sections are the solutions to gravity that is between two objects where the force is along the line between the center of mass of both objects and the strength is inversely proportional to the square of separation. Now you need polar cpordinates and the direction of gravity changes as things move around. This works pretty well for orbits around the earth and orbits around the sun, because the earth is so much more massive than sattelites that orbit it, and the sun is so much more massive than things that orbit it. If you need really precise orbit trajectories, ellipses aren’t truly accurate either. You need to account for all the orbiting bodies in the system. The 3-body problem famously doesn’t have purely analytical solutions, and you need to resort to numerical methods to calculate trajectories.

So both solutions come from simplified mathematical models. Despite being simplifications, their predictive power is actually very goood. However, like you are intuiting, it’s important to know when those simplifying assumptions lead to errors that start to become important. It’s hard to come up with a particular threshold for when you need to switch from one model to another, because it really depends on how much accuracy your application needs.