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The actress, 30, tells PEOPLE at the Chanel No.5 In The Stars event at New York City's Rockefeller Center on Friday that her famous father gave her an especially thoughtful present one year when he gifted her a necklace inspired by her cute childhood nickname.
"My dad got me a piece of jewelry that's engraved," Dylan says. "It says my nickname, which is 'Peaches' because I guess I devoured peach when I was really little and loved peaches."
Dylan tells PEOPLE her dad has used the sweet "Peaches" nickname for years, and used his excellent eye for jewelry to create the memorable piece.
"He's always called me that and he got a ring engraved and he has really amazing taste in jewelry," Dylan says. "So that was really special."
Dylan was a guest at Friday's Chanel event, a birthday celebration in honor of the 100th anniversary of the Chanel No.5 fragrance. The iconic fashion brand took over the Rockefeller Center ice rink to kick off the weeklong celebration, which will be held from Nov. 6-12.
When it comes to gifts, Dylan says dad isn't the only one picking out perfect presents. Dylan tells PEOPLE her mom, Robin Wright, once treated her to a pick-me-up treat after she went through her first breakup.
"My mom, when I was 16, you know, being an angsty teenager, was going through my first breakup and she surprised me with a trip to Napa and we did a spa day," Dylan shares.
Wright, 55, also cooked Dylan her favorite dinner to cheer her up, the actress remembers.
Getty Robin Wright and Dylan Penn
"My favorite meal is chicken pot pie," Dylan explains. "We went to this restaurant and they didn't make [it]...and she made the pot pie for me in the kitchen."
Dylan has since followed in her famous parents' footsteps by becoming an actress herself. She recently costarred in the film Flag Day alongside Sean, who also directed the movie. Dylan told PEOPLE in August that working with her father on Flag Day was a therapeutic experience.
"For me, it was equal to doing family therapy in a room full of strangers," Dylan said of filming certain scenes for the movie. "I've done family therapy, but I've never done it with a crew that I didn't know."
Dylan added that Wright inspired her to work with Sean on Flag Day.
"She very simply said, 'I've never worked with a better director than your dad. It was the most amazing experience I've ever had as an actor,'" Dylan told PEOPLE. "So I trusted that."