It's true that no dog is truly hypoallergenic- as they all have dander.
. Dander doesn't just come from fur, but it is also found in skin, urine and saliva. You could consider dander to be “pet pollen”. However there are dogs that are known for producing small amounts of dander, thus most allergy sufferers do not have reactions. However different people have different allergies. Because you are allergic to one dog, doesn’t mean you will be allergic to all dogs.
Although many Designer Breeds (such as Labradoodles) are often touted as being hypoallergenic, there is little evidence that this is so. The level of dander and shedding in mixed breeds is not consistent and can’t be predicted.
I would go to animal shelters and see what your dad can handle. Chances are you can find a yorkie or yorkie mix in a shelter that is in a dire need of a home.
I suggest using
http://www.petfinder.com/index.html
You can specify a breed and search around your local area for shelters.
Yorkies are considered as "hypoallergenic". Here's a list of "hypoallergenic dogs". Although remember, your dad may not be still be allergic to these.
American Hairless Terrier
Basenji
Bedlington Terrier
Bergamasco
Bichon Frise
Bolognese
Border Terrier
Bouvier des Flandres
Cairn Terrier
Chacy Ranior
Chinese Crested (hairless)
Coton De Tulear
Giant Schnauzer
Hairless Khala
Havanese
Irish Water Spaniel
Kerry Blue Terrier
Lagotto Romagnolo
Lowchen (Little Lion Dog)
Maltese
Miniature Poodle
Miniature Schnauzer
Peruvian Inca Orchid
Polish Owczarek Nizinny
Portuguese Water Dog
Puli (Pulik)
Shih-Tzu
Soft Coated Wheaten Terrier
Spanish Water Dog
Standard Poodle
Standard Schnauzer
Tibetan Terrier
Toy Poodle
West Highland White Terrier
Wirehaired Fox Terrier
Xoloitzcuintle
Yorkshire Terrier