Parton's death was announced by her nephew Bryan Seaver, who said sharing the news was "an honour that is mixed with absolute pride and great sadness".
Swift, 36, said in a statement shared with media outlets that "a world without Dolly doesn't feel possible, real, or right".
"But because of the eternally generous way she spent her time on this earth and the countless lives she changed for the better, her legacy will be an everlasting one," she said.
In July, Parton thanked Swift and her then-fiance Travis Kelce in an Instagram post for their $US2 million ($A2.8 million) donation to the country songstress's non-profit, Imagination Library, adding she was "blown away" by the gesture.
Swift joined a chorus of other famous faces in honouring the singer.
Paul McCartney paid tribute to the country star in a heartfelt post on Instagram, describing Parton as a "magnificent star" and "wonderful woman" who he met at the Grand Ole Opry in Nashville, Tennessee, country music's most well-known venue and where Dolly Parton first appeared at age 13.
"Her songwriting, her singing and her philanthropy were unrivaled in the world of country music and it is hard to imagine living in a world without Dolly," he said.
Beyonce, who worked with Parton on her 2024 album Cowboy Carter, said in a statement on her website: "Dolly, you were the barometer. The north star. Bold and brilliant. Talented and tenacious. You taught so many lessons on how to be uniquely oneself. I am grateful for your art, your heart, your generosity, your humor, your entrepreneurship."
Pop star Sabrina Carpenter also paid tribute to the country star, who featured on the deluxe version of her song Please Please Please.
She said in a post on Instagram: "I will cherish our laughter and conversations and I will keep your wisdom, your playful outlook on life, and perfect sense of humor in my pocket for the rainy days."
Julia Roberts paid tribute to her Steel Magnolias co-star, telling Page Six in a statement: "Dolly was a magical and joyous human. I hold knowing her as a treasure in my life. The sun has dimmed today."
Sally Field, who played M'Lynn Eatenton in the 1989 film, posted on Instagram: "If there are truly angels on earth, then we have just lost one. Dolly Parton. There will never be another. I adored her with all my heart and will always cherish our time together. Forever. Rest in peace beautiful Dolly."
The Empire State Building tower lights honoured the singer by turning the lights to a "pink sparkle" at 9.25pm, local time.
Sharing a photo of them together, actress Lily Tomlin said she had "no words", also sharing a broken heart emoji. The pair were lifelong friends, also starring in the 1980 film 9 To 5 alongside Jane Fonda.
A "devastated" Fonda posted a photo of her and Parton laughing in the set of 9 to 5 and said "Dolly will always be with us because she was so present, so powerful, so generous and playful that she is woven into all our lives and hearts forever".
Country singer and actor Billy Ray Cyrus said Parton recently messaged him to say that she "loved me with her whole heart" adding that he would "always" love her in the same way.
Parton was also the godmother to his daughter, singer Miley Cyrus and appeared alongside the father-daughter duo on their Disney show Hannah Montana.
Country singer Reba McEntire made a personal tribute to her "buddy Dolly" and wrote on Instagram: "Your love, your music and you gave so much with your huge, huge heart. There will never ever be another you."
The Muppets' official Instagram page also paid tribute, saying "she filled the world with light and we are so grateful for every second we got to spend with her onstage and off".
Rolling Stones frontman Mick Jagger also paid tribute, saying "what a wonderful songwriter and singer Dolly was. She brought so many people together, we will all miss her so much."
Bette Midler described Parton's voice as "pure gold" with a "heart to match", and mourned "a terrible, terrible loss".
Singer Dionne Warwick - who collaborated with Parton on the 2023 song Peace Like A River - posted on social media: "I will certainly miss her laughter, jokes and her being. Rest well and in total peace my friend. My heart is heavy."
Singer Billy Joel wrote on Instagram: "Dolly Parton has transcended time and become one of those iconic artists who is known for just being Dolly, beyond a specific song or moment in time. I was one of the lucky ones whose music she covered."
Celine Dion described the singer as having a "kind heart and a generous spirit" and Tom Jones said Parton's death had left a "great emptiness".
TV presenter Oprah Winfrey said one of her fondest memories with Parton was "having a hoot and a holler while wearing Bob Mackie gowns on her show Dolly".
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