As Daniel Rust notes, the definition of parallel is that two lines don't meet. What some people are trying to point out as examples are situations where lines cannot be parallel. These settings help regularize the geometry. For example, spherical geometry takes place on the surface of a sphere. The "lines" in spherical geometry are the "great circles": the circles which have the diameter of the sphere. Note then that two lines always intersect in a "point" (which in spherical geometry is defined as the two points opposite each other on the sphere).
Spherical geometry regularizes plane geometry in several ways. First, it elminates parallel lines: now every two lines intersect in a point, and every two points define a line (exercise!). Second, it unifies the treatment of lines and circles: everything is now a circle, in effect.
So "parallel" does strictly mean two lines that do not meet, but there are ways to eliminate the concept with a suitable geometry. Projective geometry is another very useful but more complex way to do this.