A plane-parallel atmosphere is a model atmosphere that ignores the curvature of the underlying body, i.e., is dealt with as if it is planar/flat. Such models are used in meteorology (Earth or other bodies) and for stellar models. They are especially suitable for (spherical) bodies where the atmosphere is a relatively thin layer around the body, i.e., "locally, virtually flat". Their use is typical in 1D models.
Perhaps the primary simplification is that when dealing with lines at an angle, i.e., neither vertical nor horizontal to the surface, the lines are treated as if the atmosphere is flat. For example, when calculating the effects of a beam of light passing through at such an angle, the light is treated as if it is passing through flat, parallel layers, any effects of the atmosphere-layer's curvature ignored.
Use of the plane-parallel atmosphere makes models easier, often making the difference in the model's practical tractability. This is true both to analytic models and computer simulations.