Wait, I don't think I ever actually made a theory post, shit.
The basis behind this is simple: Carrot uses Rokushiki, so where did she learn it from (if you watch her fights, she uses Soru and Geppo regularly):
Her paw based attacks are also similar to Shigan.
Next, just look at the characters. Oda has drawn them in very similar outfits, and introduced them in arcs right after each other.
Look at the capes, the hats, the focus on white/pink/orange.
My current theory is that Zepo, Bepo's brother, was Carrot's father, and met Stussy on Whole Cake, since we know Stussy travels there.
Pedro felt responsible for Zepo's death (since... well, he was), and raised Carrot as his own daughter.
It doesn't have much legs to stand on, but when Oda introduces Straw Hats with missing parents, it generally becomes a plot point later on. The only Straw Hat I don't expect to see ANY parental stuff from is Brook, because... yeah.