Indeed both sides of a part can be planes. If the planes are a constant distance apart over infinity then the planes are parallel. By definition a plane is flat. If the planes are not a constant distance apart then they are not parallel, although they are both flat. If the surfaces are not planes, but are a constant distance apart, then the surfaces are neither flat nor parallel.
To take Nigel’s example of two concentric circles; I agree that two points, one on each circle, and on a common radial will be the same distance apart anywhere around the circle. However, they are not parallel. The tangents to the circles at the two points will, however, be parallel.
It all boils down to basic Euclidian geometry; which I suspect is a step too far from the OPs original question!
Tel: Sadly the earth isn’t flat, but is based on spherical geometry, which is a non-Euclidian geometry in the which the 2D surface happens to be a sphere.
Regards,
Andrew