If a line in the real world is parallel to the picture plane, then its perspective image has no vanishing point.
If two or more lines in the real world are not only parallel to the picture plane, but are parallel to each other, then their perspective images remain parallel in the drawing.
In this painting, all of the superimposed green lines are extensions of lines that are horizontal, parallel to the picture plane, and parallel to the ground in real-life; notice that they remain parallel to each other in the image.
Similarly, all of the superimposed pinkish lines are extensions of lines that are vertical and perpendicular to the ground in real-life; notice that they remain parallel to each other in the image.