I would prefer a more generic statement as the heading of this thread:
Listen closely, people: parallel planes NEVER intersect.
From this it can then be derived that no line contained in a plane can EVER intersect with any line contained in another plane parallel to this plane, whether the respective lines be straight or curved.
To restrict this axiom to straight lines, you have to define a straight line as the intersection line of two planes, and parallel straight lines as the respective intersections of a plane with two parallel planes.
Spherical geometry, as far as I got it from this thread (I left school at 16), is confined to the planisphere, and I don’t know if it can be extended to concentric spheres. If yes, the same postulate as with two parallel plans would apply, i.e. two concentric planispheres NEVER intersect.
You may have noticed that parallel has been replaced by concentric. Are these planispheres parallel? No, only two infinitely small areas as defined by the respective intersection points of a radius line are parallel. Really parallel? As far as I can understand, infinitely small areas or infinitely small segments cannot be parallel because they have no orientation, only their tangents can be parallel.
The problem under discussion is purely semantic, as in the statement
“parallel lines never intersect” the term “line” is (linguistically speaking) an ellipse.
(For the non linguists, here is my own personal explanation of what a linguistic ellipse is: when you look at a circle from a point not situated on its generation axis, the circle appears as an ellipse. Therefore, if you present an ellipse to somebody, you may assume that if he is familiar with spatial geometry he may as well perceive the ellipse as a a potentially perfect circle).
Hence, in the said statement the term “line” will be interpreted as “straight line”.
Whereas in the statement: “parallel lines MAY intersect”, the elliptic term “line” will be interpreted as “curved line”.
Let’s try an analogy:
Listen closely, people: The LHC will NEVER generate a black hole.
Those who infer that “LHC” and “black hole” are elliptic expressions, may then object:
Listen closely, people: The next LHC MAY generate a financial black hole.