What "rest" is for our horses depends on what part of the season you are talking about and which horses.
During the late winter/spring legging up process for the older, experienced horses, they are generally ridden 2-3 times a week. [They are turned out in pasture 24/7.] Once the season gets going, my general rule of thumb is 1 week off for every 25 competition miles ridden to start. So an LD would get a week off, a 50 2 weeks, etc. But as the season progresses, the time off generally increases. By mid-season, the horses are doing roughly 1 ride a month and often are not ridden between rides at all.
My plan for Dream, who was out of competition most of last year and then was in jail (aka stall rest and then rehab) for a few months will involve a lot more slow legging up. Once she returns to competition, I will likely continue to ride her more frequently than I would the boys between races simply to assure myself she is fit.
For Gamer, the plan is lots of long, slow distance. That will be both as training/conditioning and as LDs to get her used to the competition game. She will get less chunks of time off as she won't be going the farther distance of the older horses. I want to get her a good solid base.