oooh, I am afraid I have no experience with reindeer. If it is parasites - assuming worms, then really that needs treating. Personally I do not like worming sick animals, as the wormers can be quite severe. But if thats causing the problem, then it needs to be done.
I have had goats really ill and given them the powder in their water and it does help. Is there anything that has changed with regard to feeding etc, that could have upset the reindeers system? Or has it eaten anything poisonous by any chance? The reindeer does need to at least drink something, as it will be dehydrated. My goats if feeling ill, will drink warm water, and even the water from soaked sugar beet, again warm.
If you managed to catch the reindeer, could you not have kept it inside for the vet to look at? I appreciate it is probably hard to catch, but really the vet needs to look to give a proper diagnosis, and could probably give it a jab, to help it feel better.
Sorry I cannot help, but fingers crossed the reindeer perks up for you.