Be careful when bringing your rabbit inside and outside all the time. The drastic temperature change between outside and inside could be to much for your bunny.
Have you tried showing them some articles on why keeping them indoors is better, instead of you just telling them? (don't get me wrong, you telling them is great, but an article wouldn't hurt)
also, Rabbits have been domesticated over the years, so them saying "He doesn't like inside he was made for the outdoors." is not all that true. Yes if he was a wild cottontail then yes, outside. but he is a domestic rabbit, which means he can live happily inside (or outside, doesn't matter)
Rabbits are social animals, meaning, they are much happier when people (or other rabbits) are around.
fur doesn't get everywhere, when they are shedding it does (they only shed twice a year) but for the most part, the fur doesn't go everywhere.
i personally think rabbits should be housed indoors, like you said, you can monitor them to tell whether they are sick or not, being housed indoors protects them from bad weather and predators, (but i see that you live in new Zealand, so that shouldn't be a problem though
) they get a lot more attention indoors as well. Not saying housed outdoors is bad, i find it be a little harder, you must make sure their place is predator proof, and that their getting enough attention and so on.
Rabbits can be litter trained (if you did not know this) you can keep your rabbit in your room (any room that your rabbit is staying in must be bunny-proofed)
or in your basement, like i do. Andy has a dog-crate that i lock him up at night, then i let him out to free range during the day (under my supervision)
also, just wondering, what breed is he?
Good luck trying to convince your parents
don't worry, it might take awhile for them to come around.