Assuming you mean Parallel in Euclidien Geometry, you'll see that condition 2 requires that "Line m is on the same plane as line l but does not intersect l (even assuming that lines extend to infinity in either direction)." i.e. if you cannot extend the two lines infinitely in either direction and still hold this property, then they aren't parallel.
I found this link too. The lines represent two blocks of time, in the discussion me and my mate had. So not euclidean geometry otherwise he'd have been correct ;)
Well that changes some things. I don't know exactly how you're defining the space you're working in of time, but you'll note that in hyperbolic space it's possible to have lines that are parallel (they never intersect in the plane) that still intersect (but only in the limit to infinity) (so as long as the two line segments don't intersect in the plane, but maybe in infinity, they'd be parallel (but only in a hyperbolic space)). So it all depends on what space you're working in.