Help with my parent's pet

[deer2art]

Ten years ago, I bought my parents a Norweigen Elkhound. She has been a joy to them. Healthy, active, absolutely loves cold weather. This winter, Gypsy began losing her hair symmetrically, all over her body and her coordination is poor. She still eats and drinks very well, uses the bathroom regularly and still loves attention but my parents have had to put her in the barn because of her fur being gone and her coordination being so bad. Outside of being wormed and shots regularly, she's never been sick or had to visit the Veterinarian.
My parents are now afraid of taking her in for fear that they will be charged with neglect because she looks so ghastly or that a Veterinarian will put her down. I told them I would see if I could find some answers. Can someone out there offer some advice on how I should handle this?