No. The circular shape makes it impossible to have parallel lines just as you cannot have parallel lines in a circle that both reach the length of the diameter of the said circle.
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None because it's a circular shape.
If by parallel, you mean two lines that do not intersect, yes, it is possible to draw them on the surface of a sphere. They will end up being circles, and most pairs will not be equal in size. If you add the idea that the two lines also continue to infinity to the definition, then you cannot draw such things on the surface of a sphere.
no parallel lines
parallel lines are diagonal lines or increasing lines
Assume there are no lines through a given point that is parallel to a given line or assume that there are many lines through a given point that are parallel to a given line. There exist a line l and a point P not on l such that either there is no line m parallel to l through P or there are two distinct lines m and n parallel to l through P.