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a trapezoid has a pair of symmetry lines but a square diamond or parallelogram all sides are parallel
A rectangle has two lines of symmetry, the lines that connect the midpoints of the parallel sides of a rectangle are lines of symmetry of the rectangle.
The shape has to be a parallelogram since it has no lines of symmetry and all sides are equal and parallel
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.
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.