I decided to finally consolidate some headcanons and theories about Mihawk’s parents. Parts that are confirmed canon are in bold. Feel free to take the rest with a grain of salt.
I think the manga has given us enough clues that Mihawk is Shakky and Rayleigh’s son, including the fact that she retired “40 years ago” and the increasing frequency Oda is showing us young Rayleigh wearing the same paisley pattern on his coat.
The only other reason I can think of that explains the paisley pattern is that the paisley is some kind of WorldStrongestSwordsman-only pattern, only worn by the best like the yellow jersey in the Tour de France, lol.
Mihawk looks quite a bit like Shakky and seems to have the same hobbies (drinking and reading the newspaper).
All of the women of Amazon Lily have flower names. Shakky (whose full name means “Peony” in Japanese) is a former empress of Amazon Lily and former captain of the Kuja Pirates from two generations before Boa Hancock.
I think the shirt Mihawk wore to Roger’s execution has peonies on it too!
Shakky joined Rocks Pirates at some point (only half-bolded because it’s not confirmed yet xD). During this time, she met Rayleigh and they fell in love. Shakky retired before the God Valley Incident, likely because she got pregnant with Mihawk.
But don’t Amazon Lily women only give birth to female babies?
I think the key is that only female children are born on the island. The sex of the infants born on Amazon Lily might be due to an environmental factor that affects the island itself. It might be dietary, or a magnetic field, or perhaps the love sickness can affect a child’s sex, which would explain why the sickness is so taboo!
An extremely rare male Kuja infant could be born with features that we don’t see often, such as red eyes or hawk-vision! Mihawk’s eyes could also be inherited from his maternal grandpa, who we know nothing about. That is, if they’re a heritable trait at all; I tend to believe they’re just a design choice by Oda to show Mihawk’s intensity.