Not necessarily. All parallelograms are quadrilaterals but not all quadrilaterals are parallelograms. Parallelograms are more specific instances of quadrilaterals. Its like saying all squares are rectangles but not all rectangles are squares.
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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.
It's not a matter of 'finding' it. Either the quadrilateral is a parallelogram,or else it is not one.If the opposite sides of the quadrilateral are parallel, then it's a parallelogram.
A quadrilateral is a parallelogram if it has two pairs of (opposite) sides that are parallel.
The second statement is false.