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Infinitely many. Polygons with 6 or more sides can have 3 pairs of parallel lines. Polygons with 7 or more sides can have a set of three parallel lines.
You can construct any polygon with more than 4 sides such that two sides are parallel. A quadrilateral in which two lines are parallel is called a trapezoid.
There are more than 4 other types of quadrilaterals.A trapezoid has only one pair of parallel sides. All other quadrilaterals have no parallel sides or two pairs.
Any polygon with more than 3 sides can have two pairs of parallel sides.If it has only four sides and they make two parallel pairs, then it's a parallelogram.
In space with 3 or more dimensions, there are infinitely many pairs of lines that are not parallel and do not intersect.