A hexagon or any of a selection of polyhedrons with 10 or more sides. I am not sure that less than 10 sides will suffice but they may well do.
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A 4-sided polygon with four or three sides of different lengths can readily be constructed so that no two sides are parallel. The shapes so obtained will not have an axis of symmetry. However, a kite is a quadrilateral having two pairs of adjacent sides of equal length but does not have any parallel sides. It is symmetrical about its long axis.
No polygon can have less than three sides.
A polygon with three sides is a triangle, the prefix tri- meaning three.
No, a parallelogram cannot be a hexagon. A parallelogram is a quadrilateral with opposite sides that are parallel and equal in length, while a hexagon is a polygon with six sides. These are two different shapes with different numbers of sides.
A polygon must have at least three sides.