The advantage of the "Parallel Push" is the _direction_ the pads move in.

Simple cantilevers, either cable style, move in a circular arc centered on the frame/fork boss.

For bikes with skinny rims and fat tires, this creates difficulty because the brake shoe can rub on the overhanging, bulging tire's sidewall when it's in the rest position. This is especially probematical if the brake bosses are widely spaced. The brake shoes move downward as they move inward, to a greater or lesser extent.

If you lower the brake shoe to safely clear the tire, it risks "diving" under the rim as the brake shoes wear.

The "Parallel Push" linkage causes the brake shoes to be traveling in a roughly horizontal direction in their normal range of motion. (Actually, it's still a circular arc, but it's centered on the lower, inboard pivot of the parallelogram.)

Sheldon "Geometry" Brown