The definition of a line of symmetry is a line that can be draw down the center of any shape or object to show mirror image of the other side. where each side is a mirror image of the other side. Now for a horizontal one, it just means that if you cut the figure in half, horizontally it will make a mirror image.
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By definition, lines are parallel if they have the same gradient (slope). Any horizontal line has a gradient of 0, so it is parallel to any other horizontal line.
The letters S and N have point symmetry but not line symmetry.
The Japanese flag has four lines of symmetry. Two of them are a horizontal line halfway up the flag, and a vertical one halfway across the width. It also has two rotation lines of symmetry across the diagonals.
A parallelogram does not have a line of symmetry.
draw line of symmetry for 20