Are you saying that the only time they're allowed out at all is during those time periods, and that during those times are the only time you are allowed to ride them? That seems a bit... extreme to protect against the heat, especially if a horse is in training.
My horses are on pasture board, and are out all day and all night with a couple other horses. At the barn, though, they keep the horses in with fans on all day and let them into another pasture at night (and swap during the winter, minus the fans). But that doesn't mean that the horse's owner can't come during the day to train with their horse, or to even go on a pleasure ride. Everyone is free to come and ride their horse when ever they want during the day.
I suppose we're all wanting further explanation so we can better understand your personal situation and can give better advice.