I agree with everyone saying that this is an awful situation for a bunny to be stuck in and shame on your friend’s parents for creating it. Still, I think it’s great that your friend (and you!) care enough to recognize the problems and want to help. If her parents don’t come around soon, I do think the bunny would be better off elsewhere (shelter or another home), but in the meantime here are a few suggestions for “bandaids” to improve the bunny’s life a little bit:
DIET: Besides carrots, what does the bunny eat? If pellets, make sure he’s getting high quality pellets. Timothy-based pellets are especially important for bunnies that don’t eat a lot of hay for whatever reason. Also, maybe grandma could try feeding him something like lettuce or veggies instead of carrots? It’s just as fun to watch a bunny gobble down big pieces of greens/herbs. And as far as hay, keep trying! See if your friend can get ahold of different kinds of hay (timothy, orchard, oat, etc) in case the bunny finds one he likes. Even grass is better than no hay! Try reducing the amount of pellets to see if he’s more receptive to hay when he’s a little hungry.
LIVING SPACE: Like Sarah said, a cardboard box with a small opening makes a great hidey hole. It’s easy and cheap to make, and even your friend’s parents couldn’t complain. The cage itself is definitely too small to live in – could your friend bring the bunny into the house to play in her room every day? In fact, what about just putting the cage in her room? That way her parents wouldn’t have to move their furniture. If the cage is too big, the bunny could just live in her bedroom with no cage as long as it’s bunny proofed (he’d have to be litter trained). Or at the very least, could your friend buy an x-pen or similar kind of fence to let the bunny run around outside (supervised) for a little while each day? X-pens can be easily moved so your friend could put it away when not in use so it’s not an eyesore. Does the cage have a wire floor? If so, put something hard on the bottom (a sheet of wood, some tiles, a piece of plastic – whatever) to protect his feet.
GROOMING: Bam had a great suggestion about using cornstarch to clean the bunny’s butt, plus baby wipes (or a wet washcloth) can work well especially if the poop is still wet. Trim out what you can (carefully!). Make sure to clean the cage as often as possible so the bunny is never sitting in poop/urine. Hopefully the consistency of his poop will improve if his diet gets a little better… Brush him regularly to keep his fur from getting matted, especially near his butt because dirty fur mats easily. Has she ever tried trimming his nails?
I don’t know much about your friend’s situation so those are all the ideas I have for now. I’m not sure what the climate is like where you live, but be aware that bunnies have a low tolerance for heat and can’t always survive outdoors year-round. Is your friend old enough to save up money and/or drive to the vet herself? I do think a vet visit is important, particularly with the diarrhea and no-hay diet – he might not be eating hay because his mouth hurts and his teeth are probably overgrown and in need of clipping.
Good luck! Your friend sounds like a caring person and I’m sure that in the future when she’s independent, she would make an excellent bunny mom if she chooses.