It probably comes from something related to horses before cars, but I'm not sure. You say "pull over" to drive to the side and stop. Pull up and drive up can have the same meaning though.

That "lace" is called "the reins." It's pronounced exactly like "rain"+s. Yes, "pull up" the reins of a horse to signal it to slow down or stop. It's a standard term in horsemanship. As a matter of fact, using Google Books I just found an 1877 book, "Dictionary of Americanisms: A Glossary of Words and Phrases Usually Regarded as Peculiar to the United States," which says: "To pull up, among travellers, means to stop. Alluding to the act of pulling the reins of a horse, in order to stop him."

There are many phrases that can be used for stopping. Pulling up is typically only used for cars and other vehicles, and other terms related to these are pulling in, pulling out, and pulling over. Pull in, to drive away from the centre of the road towards a desination: kerb, garage, car park Pull over, to drive away from the centre of the road, possibly to do so and then stop. Pull out, to drive toward the centre of the road, or to start to drive from a parking position onto the road, or to drive from a side-road onto the main road (etc) I'm also not sure of the origin, but it should have something to do with forces (maybe originating from horses) but to pull always means that a force has been exerted onto it: (of an animal or vehicle) be attached to the front and be the source of forward movement of (a vehicle). "the carriage was pulled by four horses" (of an engine) exert propulsive force; deliver power. "the engine warmed up quickly and pulled well" If you go back to the original word of pull, it comes from Old English: PULLIAN ‘pluck, snatch’; origin uncertain; the sense has developed from expressing a short sharp action to one of sustained force. So if you're forcing a car or horse to do something, you're essentially 'pulling' it into another direction. hope this helps! :)