Is two distinct rhumb lines with the same bearing on a perfect sphere parallel?

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Is two distinct rhumb lines with the same bearing on a perfect sphere parallel?

I am defining parallel as 2 objects as parallel if they do not intersect, go in the same direction, and a constant distance can be defined between the 2.

(Note: A rhumb line, or loxodrome, is a “line” crossing all meridians at the same angle, i.e. a path of constant bearing. It is obviously easier to manually steer than the constantly changing heading of the shorter great circle route.

I am having a hard time imagining it. How do I show or explain if it would be parallel or not?