That is %¤#&!§-. You don't have to get rid of your dog and I find it unlikely that you got your parents sick because of your dog's parasites. Maybe your parents are just allergic to dogs and you having dog hair or dog dander on you made them sick. I would never get rid of a dog, especially not for a stupid reason. Personally I'd choose my dog over my parents.. The dog would never make me make a choice like that. They sound like dog haters in general to automatically assume its the dog.
Take your dog to a holistic vet and use Dr.Clark's parasite cleansing program if it makes you feel better about getting her checked out. Or tell them you did get rid of her and then go up again a few months later and see if they're still "sick".