Not normally on its surface.
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Yes, in plane geometry parallel lines continue forever. However, in polar geometry (3 dimensions, as in Earth longitudinal lines), parallel lines eventually intersect at the poles of the sphere,
In Euclidian geometry, which is the geometry of a plane surface, parallel lines do not intersect because that is the definition of parallel lines. But note that there are other geometrical systems in which parallel lines do intersect, for example if they are drawn on the surface of a sphere. Definition of parallel lines: Lines that always stay the same distance apart and never meet.
For most purposes in algebra and geometry, but especially geometry, parallel lines never meet. This should be the answer you give on nearly every question. However, speaking realistically, parallel lines can meet on planes of negative and positive curvature. An example of positive curvature would be a sphere; on a sphere, if you try to draw a triangle, the interior sum would be more than 180degrees and parallel lines would intersect. Similarly, on a plane of negative curvature like that of a surface of a saddle, the sum of the measures of the triangle would be less that 180 degrees and once again parallel lines will intersect.
the answer is parallel lines
parallel lines are slanted lines