What do parallel lines look like on the pseudosphere?

As an application of Clairaut's Relation, you can work out quite explicitly what the geodesics on the pseudosphere are. The most useful form can be found in formula (6) on p. 258 of doCarmo's Differential Geometry of Curves and Surfaces or in exercise 23 on p. 78 of my differential geometry text. In particular, other than the meridians (the copies of the tractrix), we have curves that turn around the pseudosphere from the fat end to a parallel circle, hitting it tangentially. Choosing constants carefully, you can see that infinitely many of these will pass through a given point and miss a fixed meridian.

Any of the "vertical" lines in this image of the pseudosphere are parallel to each other.