It may be.
A quadrilateral has four sides, but they need not be all the same length, nor parallel to each other.
It's true that a parallelogram *is* a quadrilateral.
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The key is the object is quadrilateral with parallel sides. It is a parallelogram. It is also a rhombus.
A rectangle is a type of quadrilateral, and it also is a parallelogram.
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No. A parallelogram is a special kind of quadrilateral.
A shape that is a parallelogram but not a quadrilateral doesn't exist.