Not true at all, they just need to be at the Non-Euclidian Ball at the same time

What if we make the parallel line a perpendicular one in the shape of a kitten? Surely this two parallel lines will meet.

"As we have matured, our tastes have changed and we've grown apart intellectually, philosophically, politically, and in our recreational and culinary interests and now fall along parallel lines which will never again intersect"
- Ray Stevens

They either never meet, or they meet everywhere