Naruto's relationship with the other Jinchuriki would likely change. If Naruto hates Yondaime, he might see the system of creating Jinchuriki in a more negative light. He could try to convince the other Jinchuriki that they have been wronged by the villages and their leaders like Yondaime. This could lead to either a united front among the Jinchuriki against the villages or some internal conflict among them depending on their individual beliefs.
Well, in the original, Yondaime was Naruto's father and a hero. But in this fanfiction, if Naruto hates him, it completely flips the relationship. It makes Yondaime more of a villain - like figure in Naruto's eyes.
His relationship with Minato would be very different. Instead of a father - son bond, it would be more of a brother - in - law type of relationship. Minato might see Naruto as an equal in a different way, and they could collaborate more as peers. Also, Naruto's relationship with Sasuke would have a new angle. Sasuke might be more envious of Naruto's strong Uzumaki bloodline, which could intensify their rivalry.
Perhaps Naruto blames Yondaime for his mother's death. In the original story, Kushina died while giving birth to Naruto, and if in this fanfiction there are some hidden motives or actions of Yondaime related to that event, Naruto could hold a grudge. Like if he found out that Yondaime could have saved Kushina but chose not to for some greater good that Naruto doesn't understand or accept.
For Naruto and his friends, his new connection to the Dragon Sage power from Kushina could make some of his friends a bit jealous at first. However, as they see how he uses it for the good of the village, they would come to accept it. His relationship with Sasuke could also be affected. Sasuke might be intrigued by this new power and see it as a way to also become stronger, which could lead to some friendly competition between them. With Kushina, Naruto would share a deeper emotional connection as they explore their shared power heritage together.
If Kushina was alive in a fanfiction, Naruto's relationship with his father, Minato, would be different. They could all be a proper family, and Naruto would grow up with both his parents. His relationship with his friends might also change as he would have a more stable home life. For example, he might be more confident around Sakura and Sasuke, and his bond with them could be less about seeking acceptance due to his lack of family.
If they are revived, Naruto's relationship with his parents would be much more complex. He has grown up without them, and now he has to adjust to having them around. Minato and Kushina would also have to get used to the fact that Naruto is no longer the little boy they left behind. Their relationship would be filled with a mix of catching up and learning about each other all over again.
His relationship with Sasuke would likely be very different. Since Sasuke has also had experiences with the Rinnegan, they might now have a new level of understanding between them. They could train together to master the powers of the Rinnegan or compare notes on how to use its abilities.
One possible reason could be a misinformation plot. Maybe in the fanfiction, Naruto has been led to believe that Yondaime had some dark secrets or did something bad to his family that he discovers later in the story.
With Sasuke, Naruto's parents being alive could lead to a more positive influence on Sasuke. Naruto might be more stable and happy, and this could rub off on Sasuke. They could become closer friends earlier, and Naruto could help Sasuke deal with his family issues in a different way. Maybe Naruto's family could even offer some support to Sasuke after the Uchiha massacre didn't happen.
His relationship with Sasuke might be different. With Kurama's influence, Naruto could be more confident in reaching out to Sasuke. He might be able to understand Sasuke's pain better and be more effective in bringing him back to the village.