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yes it is a symmetrical shape
A shape that is a parallelogram but not a quadrilateral doesn't exist.
Yes, a parallelogram has 2 pairs of symmetrical sides. A rectangle is an example of a parallelogram but with all of its angles being 90 degrees whereas in a typical parallelogram none of the angles are 90 degrees. Opposite sides will be of equal lengths.
A shape must have 4 sides in order to be a parallelogram (minus other defining characteristics a shape must have to be a parallelogram).
Asymmetry is when a shape is not symmetrical.