In James Anderson's Hyperbolic Geometry, two lines in the upper half-plane model of the hyperbolic plane are said to be parallel if they are disjoint.
Suppose $l_1$ is the half-circle centered at $(0, 0)$ with radius $1$, and $l_2$ is the half-circle centered at $(2, 0)$ with radius 1. These circles intersect at $(1, 0)$, but this point is not a part of the upper half-plane. Would it be right to say that $l_1$ and $l_2$ are parallel?
Terminology does vary a little, so I won't use the word parallel. I do not know the book you mention.
One thing that is special about your lines is that there is no common perpendicular line to them. If you begin with a line segment and two lines orthogonal to it at both endpoints, we know that these lines cannot meet, as that would create a triangle with angle sum at least $\pi.$