Dolly the dog, named in honour of the star, was found in a tent during the clean-up of the festival in Somerset.
The white Lurcher has been housed at the Happy Landings animal shelter.
68-year-old Parton sent a video message saying she was "very honoured and flattered" that the dog had been named after her, the BBC reported.
The singer-songwriter, multi-instrumentalist, actress, author and philanthropist said: "I had my manager call the Happy Landings animal shelter to make sure the dog is being treated and cared for properly.
In a message on its website, Happy Landings said Dolly the dog is a "sweet-natured older lady" who arrived with a "serious ear infection".
The shelter said it had received numerous phone calls from people worried about the dog, after her story came out.
Parton has already had another famous animal named after her.
Dolly the sheep, the first successfully cloned mammal, was named for the star after doctors revealed the animal was copied from a mammary gland cell.
The country singer received glowing reviews after her performance at Glastonbury, having attracted one of the biggest crowds of the weekend to her set.
Glastonbury Festival is a five-day music festival that takes place near Pilton, Somerset, England.
