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A parallelogram has no symmetries. You can divide it in half, but it has no symmetries. However, it has lines which divide it in half. There are an infinite amount of them.
My point is, A PARALLELOGRAM HAS NO SYMMETRIES.
~ Abigale Kim (a+bi)
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While parallelograms have no reflective symmetries, (there is no line that is an axis of symmetry in a parallelogram), parallelograms do have central symmetry.
This means when you rotate it 180 degrees about the center, it stays the same. You can think of reflective symmetry as the graph of an even function and central symmetry as the graph of an odd function.