The white lurcher was discovered abandoned in a tent during the festival clean-up last week.
She was taken to the nearby Happy Landings animal shelter, where she was dubbed Dolly in honour of the Jolene hitmaker, who was the star turn of the 2014 event.
Parton has now offered to adopt the dog if its owner does not claim her.
She said in a video message: "I had my manager call the Happy Landings animal shelter to make sure the dog is being treated and cared for properly.
"At this time, nobody has claimed the dog and the dog is in great hands at the shelter. I will take the dog home to America if nobody claims her within a reasonable amount of time."