distance - In relativity, what is the difference between a rod that is perpendicular to direction of motion and a rod parallel to the direction of motion?

In Feynman's Lectures on Physics , chapter 15, page 6 he writes about 2 identical, synchronized light signal clocks. These are clocks that consist of a rod (meter stick) with a mirror at each end, and the light goes up and down between the mirrors, making a ,click each time it goes down. He describes giving one of these clocks to a man flying out in space, while the other remains stationary. The man in the space ship mounts the clock perpendicular to the motion of the spaceship. Feynman then writes:

"the length of the rod will not change. How do we know that perpendicular lengths do not change? The men can agree to make marks on each other's y-meter stick as they pass each other. By symmetry, the two marks must come a the same y- and y' coordinates, since otherwise, when they get together to compare results, one mark will be above or below the other, and so we could tell who was really moving."

what exactly is the "test" with the marking of meter-sticks that Feynman is describing? why would it violate relativity, since it seems the person in the space ship would be looking outside? why would a change in a perpendicular length violate relativity, but not a change in parallel length--couldn't the men also make marks on each other's sticks, in the case of parallel lengths? Thank you!