The lovelorn ballad only reached number 108, but it set the "dirt poor" Tennessee singer on the path to fame.
Since then, she's sold more than 100 million albums, topped the country chart 25 times, and won eight Grammy Awards - but she shows no signs of slowing down.
"Almost every day I come up with a few song titles or a sweet melody," says the 70-year-old, who claims to have written more than 3,000 songs in her career.
Her latest album, Pure & Simple, came out on Friday, featuring new songs with stripped-down arrangements that recall the early days of her career.
It has an overwhelmingly romantic theme, inspired by renewing her wedding vows earlier this year. And just like her chart career, her marriage to Carl Dean turned 50 this year.
"I thought, well, this would be a good year to write an album of love songs," she says. "I've never done that before. I've usually had a story song or a gospel song, but this was all love stories."
The BBC caught up with the star to discuss the new record, her legendary Glastonbury set, and why she admires Adele.
First of all, congratulations on celebrating your golden wedding anniversary!
Thank you so much. That's a long time to do anything, ain't it? Especially to be married.
What's the secret?
We've always been good buddies. We have a lot of fun and a lot of respect for one another. It was his first marriage and mine and we never thought we'd ever want to do that again. Why bother?
You renewed your vows earlier this year. What was that like?
That was sweet. There was no pressure at all. We had our own little ceremony in a little chapel on our property, then we went in our little RV down to Ringgold, Georgia, and spent the night where we had married 50 years before. We took some beautiful pictures and got all dressed up. It was fun really.
Did you wear your original wedding dress? Could you still fit into it?
Well, I could but I don't know where it is now! I only had a simple little white dress that my mother had made for me for that wedding, but I had always wanted a beautiful, long, beaded wedding gown, so I dressed completely to the nines. I made a right nice bride!
That sense of romance permeates your new album.
Oh yes! In fact a lot of the songs were written just because of that.
You've written so many love songs over the years - how do you find new ways to talk about it?
Love is always growing, and we grow as people as well, so you just automatically take on new twists and turns. And it's amazing. I've always been amazed by that myself, how people can just write, year in and year out, about love and it all is different. I could write one right now!
How often do you write?
I write something almost every single day unless I'm sick or on a movie set.
Is that ever an inconvenience? Do you ever find yourself going, "I don't have time to write a song right now, I need to finish this casserole?"
Well, no! In fact, I write a lot when I cook. If my cooking's really good that day, my songs are really good.
In Head Over High Heels, you sing about "painting my eyes like Adele." What made you write about her?
Everyone loves Adele. With all my little nieces, it's all "Adele this" and "Adele that!" And I love how she does her make-up, and so I'm always saying to my little nieces, "can you fix my eyes like Adele?"
When I started that song, I had not planned to write that line. I was just thinking, "This needs to be about a girl dressing up sexy, so she's got to have some make-up, and she's got to paint her eyes". And all of a sudden it's like, "And my eyes like Adele!"
What do you think her reaction will be?
I think she'll have to be complimented, just like I'm always complimented when people write songs about me!
I just meant I wanted to look like Adele. I hope she thinks it's cute.
You both like to talk to your audience in between songs, but she swears a lot more than youâ€¦ I wondered what you made of that?
Oh, I don't pay any attention to that! Words are just an exchange, like money. I don't think the Lord holds that against us. I think that's just the way we communicate. And some people are just born cussers.
What's the worst thing you've ever said on stage?
Oh, I've never said anything real bad. I try to respect the audience.
The song I'm 16 is based on one of your sisters, is that right?
Yeah, she had a couple of bad relationships and she thought it wasn't meant for her to have true love. Then all of a sudden, in her 60s, she met someone and they fell in love, and it's just like they were born again. It's just the sweetest thing.
Another one of my sisters and I were laughing. We said, "Look at her, she thinks she's 16!" and all of a sudden I heard the song in my head.
Goes to show you're never old unless you choose to be.
What age do you think of yourself as?
I'm 16 in some ways. I'm still a hopeless romantic! But I'm 35 in my spirit and in my mind. When I was 35, it was a pinnacle, a great time in my life - success and happiness and all that. And so I just decided "I'm gonna claim that number and always be that in my state of mind."
What inspired Can't Be That Wrong?
That was an easy one to write. I think it's the ultimate cheating song.
I based it on someone very, very close to me, who was very much in love with the person at home, and their children, and they weren't looking for love, they just happened to fall for someone. They were so torn, because they loved both people. I saw the turmoil and the heartache, and how overwhelming and overpowering it was. And I just thought, "That's a strong-ass love, right there". You know, to be willing to go through all you have to, and put the other people through that. So I thought, "wow, this would make a great song."
The last time I saw you on stage in the UK was at Glastonbury - and you've put that concert on this album as a bonus disc. What are your memories of it?
That'll be one of my greatest memories, actually. That'll be a real highlight in my career.
I wasn't even certain about doing that show, because I didn't know if I was the type of artist that would fit in that type of a festival - because people are rowdy, they like to go dancing, and they like to drink and smoke and I thought, "I wonder if they're even going to pay attention?"
And so, as soon as I went out and I saw the crowd and heard the roar, I thought, "Ohhhh, my Lord, how did I miss this all these years?"
They call that part of the day the "legends slot". If you were booking next year's Glastonbury, who would you put in there?
I don't know. I hadn't even thought about it in those terms. Probably Adele.
But I'm not sure I'd do it again, because I don't know how I could ever top that.
Pure and Simple is out now on RCA Records.
Follow us on Twitter @BBCNewsEnts, on Instagram at bbcnewsents, or email entertainment.news@bbc.co.uk.
It is 50 years since Dolly Parton first entered the US charts, with a cover of The Tune Weavers' Happy, Happy Birthday Baby.