None because it's a circular shape.
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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.
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.
how many sets of parallel lines are in a triangle
2 pairs of parallel lines
There are 2 sets of parallel lines.