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.
Any polygon with 4 or more sides can have 2 parallel sides.
Any regular polygon with an even number of sides must have parallel sides.
Squares and rectangles
Quadrilaterals. Parallelograms have two pairs of parallel sides.
All regular polygons with an even number of sides. Irregular polygons with an odd number of sides can have parallel sides. There are also non polygonal shapes that can have parallel sides.
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.
Any polygon with 4 or more sides can have 2 parallel sides.
Any regular polygon with an even number of sides must have parallel sides.
A trapezium or parallelogram. A pentagon can have a pair of parallel sides. Polygons with 6 or more sides can have two (or more) pairs of parallel sides.
Squares and rectangles
No, however, polygons with an even number of sides have parallels.
Quadrilaterals. Parallelograms have two pairs of parallel sides.
There can be no convex polygon with three parallel sides. There are concave polygons with 6 or more sides that have three parallel lines.
Infinitely many. Every regular polygon with 2n sides has n pairs of parallel sides. Then there are irregular polygons which can have parallel sides. And there are also irregular shapes which have parallel sides.
Circles, ellipses, triangles, most polygons with 4 or more sides. In general, parallel sides are very much the exception.
Rectangle, Square,