If a parallelogram has a line of symmetry through each pair of opposite sides, then we have a rectangle.
But if a parallelogram has a line of symmetry through each pair of opposite vertices, then we have a rhombus.
A square is a combination of both cases above.
The statement "A parallelogram which has a line of symmetry is a rectangle" is sometimes true. Overall its false because it implies that all parallelograms with this property are automatically a rectangle.
