I posited this question to my geometry teacher in highschool many years ago, and it stumped her. I've recently brought it up again in conversations with friends and have not gotten any answer that's satisfactory. I have an intuitive feeling of what I expect, but I don't know enough to really back it up with solid reasoning.
Basically, consider this theoretical situation: Suppose you have two infinitely long lines (you could say a couple laser beams which never diffuse, but any perfectly straight line would work), and they are mounted so that they intersect at some point along an infinite plane in space. Then, you gradually rotate one of them so that the angle of intersection gets gradually closer and closer to 0, moving the lines closer and closer to parallel. At some point, intuition tells us that there is a position at which these lines will be perfectly parallel and never intersect. However, math tells us that they will always intersect, and the angle at which they intersect will simply approach 0 forever but never reach it.
Theoretically, parallel exists. If these lines were angled away from each other, they will not intersect, because at each increment, the distance between the two lines increases. If one of the lines in this case were rotated closer and closer to parallel, eventually they will angle towards each other and intersect at some distance away.
Math, again, tells us that this intersection point is found by a limit calculation, but it evaluates to infinity.
Considering that theoretical parallel does intuitively exist, when the lines become perfectly parallel, what happens at the point of intersection? There couldn't be a final point of intersection, could there?
EDIT I am de-emphasizing the following paragraph as it confuses the question, more than anything. It addresses a physical world representation where this question is largely theoretical, and concerns the mathematics behind the concept.
The only way I can wrap my brain around this situation, in my relatively limited understanding of physics and mathematics, is if these lines physically manifested as infinitely large loops, and then behave like intersecting rings. At any frame of reference, there is a point which appears parallel (the tangent lines) but where elsewhere, on the other side of the ring, they intersect. But this then assumes that the universe is curved along itself. Is there a better theory to explain this? I'm sure I can't be the first one to imagine this kind of situation.