There's a lot to know about Yū Yū Hakusho's Kazuma Kuwabara: he helps Yusuke and the others with Spirit Detective work with his powerful Spirit Sword that can slash through dimensions; he embodies the standard Japanese delinquent but also strives for academic achievement; he has a strong sense of both honor and masculinity; he hates Hiei with a passion yet loves his sister Yukina with an equal passion. This sounds like a lot, but there's at least one place where Kuwabara is sorely lacking in development-- his family life.
Most people know very little about Kuwabara's parents, mainly because they almost never show up in the series. There is a brief appearance by his father in the manga and his mother doesn't show up at all. They are also both missing from the original anime. Kuwabara is mainly watched over by his older sister Shizuru. For how little Kuwabara's parents are brought up, fans are left to infer many things about them based on what little information is given.
The best place to start might be Kuwabara's father. Where Kazuma resembled a Japanese delinquent, his father looked more like a member of the Yakuza. He completes his look with a ponytail, a pair of shades, ear piercings, a gold chain necklace, a cigarette, and a Clark Gable mustache. He's also 194cm which is apparently more important to know than his actual name.
Kuwabara's father only showed up for one scene in the final volume of the original manga, but it was a fairly meaningful appearance. It turned out he knew a lot about the goings-on between the Human and Demon Worlds. This was likely because Kazuma or Shizuru or one of their friends told them about it. In fact, the whole reason this came up was that Yusuke arranged for Yukina to stay with the Kuwabara family upon Genkai's passing. Regardless, the casual and even crass approach Kuwabara's father took to the situation is still surprising.
It's hard to say just how much Kuwabara's father knew about spirits and demons and psychics, but he was able to give some valuable insight. He noted that multicultural understanding and communication would be key in forming proper relationships between humans and demons. He also noted that Yukina, as a demon, would be susceptible to prejudice and tells Kazuma that it will be his responsibility to stand up for her. Spirit Detective or not, Kuwabara's father is an adult with a lot of life experience, so he knows how to give his children advice when they need it.
As for Kuwabara's mother, she doesn't appear on-screen in either the anime or the manga. She's only brought up occasionally and it's never anything of significance. With so little information about her, fans must draw their own conclusions about what she's like.
It's possible that Shizuru can help in determining what Mrs. Kuwabara is like. As Kazuma seems to take after their father, Shizuru might get more of her looks and personality from their mother. This can't be confirmed, but if it's true, then Mrs. Kuwabra might practice the same brand of tough love as her daughter, something not seen in the easygoing father.
While neither Mr. nor Mrs. Kuwabara is integral to the plot of Yū Yū Hakusho, knowing more about them might be nice. A child's upbringing can have a great influence on their lives from their personality traits to the choices they make. Some of these things are passed down genetically while others are conditioned by proximity to a parent. If the same holds true for Kazuma and Shizuru, then learning about their parents could inform their actions throughout the series.