I know that because the pseudosphere is a model for hyperbolic geometry, there are infinitely many lines that can be drawn parallel to another given one through an external point.
However I'm struggling to figure out a pictorial representation for this.
On
Sketching hyperbolic geometry geodesic parallel lines as I understand them.
Two red asymptotic lines through P parallel to a blue line R from among infinitely many parallels are shown here again from Wiki Hyp. Geometry
The situation in the plane can be seen here in the Poincaré disk model. From out of many parallel lines a pair among them passing through P is marked red and either of them is parallel to to the blue line R. In the plane also Poincaré model also such red line pairs parallel to a common parallel (not drawn) can be viewed.
In 3 space also I have shown a pair of red asymptotic lines through P parallel to the axis of symmetry R of a pseudosphere R.
Beltrami had shown and validated asymptotic lines of constant Gauss curvature $K$ pseudosphere as geodesic parallel lines (infinitely many parallel) pairs in three dimensional hyperbolic geometry. The parallel lines constitute a Chebyshev Net defined by Sine-Gordon equation with angle $2 \psi$ between pair of lines $$ \psi^{''} = K \sin \psi \cos \psi $$
Imho geodesic lines defined by Clairaut's Constant $( r \sin \psi$) on surfaces of revolution do not constitute parallel line sets in hyperbolic geometry. They are geodesics in the other elliptic geometry categorized by Riemann.
On
For pictorial representation: Two red asymptotic lines through P parallel to a blue line R from among infinitely many parallels are shown here again from Wiki Hyp. Geometry
The situation in the plane is given here from the Poincaré disk model. From out of many parallel lines a pair among them passing through P is marked red and either of them is parallel to to the blue line R. In the plane also Poincaré model also such red line pairs parallel to a common parallel (not drawn) can be viewed.
In 3 space also I have shown a pair of red asymptotic lines through P parallel to the axis of symmetry R of a pseudosphere R.
Beltrami had shown and validated asymptotic lines of constant Gauss curvature $K$ pseudosphere as geodesic parallel lines (infinitely many parallel) pairs in three dimensional hyperbolic geometry. The parallel lines constitute a Chebyshev Net defined by Sine-Gordon equation with angle $2 \psi$ between pair of shown asymptotic lines:
$$ \dfrac{d^{2}\psi}{ds^2} = K \sin \psi \cos \psi $$
These two unique hyperbolic parallels can be drawn on the central Beltrami and cone type Pseudospheres, constant negative K Kuen surfaces, Breather surfaces also.
Imho geodesic lines defined by Clairaut's Constant $( r \sin \psi$) on surfaces of revolution do not constitute unique parallel line as in case of hyperbolic geometry.
They are a set of many geodesics in elliptic geometry classified by Bernhard Riemann.
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.