To me the statement "two line segments are parallel if they are parts of parallel lines" is clear. How can we identify two different regions are parallel? Is it the statement "two regions are parallel if the planes in which they lie do not intersect" correct? Or when we say "two different regions are parallel" we mean intuitively the two regions are being an equal distance apart everywhere? For example, we often say "the bases of a right circular cylinder are parallel".
Two planes are parallel if every line in them is parallel to some other line in the other plane and vice versa. In $\Bbb R^3$, this amounts to the planes never intersecting (or being the same). I don't really like the distance definition because we then have to define what we mean by distance between a point and a plane, etc.
Two different planar regions can be called parallel if it's normals are parallel.