One great fanfic could be 'The New Beginning for Dark Naruto'. In this, Naruto's parents being alive changes his whole outlook. Instead of the lonely and somewhat bitter Naruto we know, he grows up with the love and support of his parents. His father, Minato, trains him personally, and Kushina is always there to give him emotional support. This alters the course of his journey in a positive way, making him less likely to turn to the 'dark' side as he has a strong family unit.
The plot could start with Naruto growing up in a loving family environment. His parents, Minato and Kushina, are alive and they are aware of the Nine - Tailed Fox inside Naruto. They train him from a young age to control the fox's power. However, the village might still be a bit wary of Naruto because of the fox. This could lead to Naruto feeling a sense of isolation from the villagers despite his family's love. As he grows, he might start to resent the village's attitude and begin to show darker tendencies as he tries to prove himself in a different way than in the original story.
The story would be very different. Naruto's parents being alive would mean he has a support system from the start. His father could pass on his jutsu and wisdom more directly. Naruto might not have the same drive to become Hokage in the same way, as he wouldn't be seeking that family - like recognition from the village. His mother's presence would also add an element of warmth and stability to his life. And perhaps he would be more of a prodigy with both his parents' guidance, changing the power dynamics in the ninja world.
His relationship with his parents would be loving. Minato would be a strict but fair father, teaching Naruto about ninjutsu and the responsibilities of being a shinobi. Kushina would be the warm, affectionate mother, always there to comfort and encourage Naruto. They would support his dreams and be proud of his every achievement.
The plot could start with Naruto being born blind. His parents, Minato and Kushina, are determined to raise him as a strong ninja. They might focus on training his other senses from a young age. As Naruto grows, he faces challenges in the ninja academy due to his blindness but overcomes them with the help of his parents and his own determination. He could develop unique fighting styles that rely on his hearing, smell, and touch. For example, he might be able to sense an opponent's chakra flow better than others, which gives him an edge in battles.
Well, one possibility is that Dark Naruto would be overjoyed. After all, Kushina is his mother. But at the same time, he could be very conflicted. His dark side might make him suspicious of her motives. He could test her in various ways to see if she's really who she says she is. And Kushina, on her part, would probably be heartbroken to see what her son has become.
One possible storyline could be that with his parents alive, Dark Naruto might not have turned so dark in the first place. His parents' love and guidance could have steered him away from the path of darkness. For example, Minato, being a powerful ninja, could have trained Naruto properly from a young age, and Kushina could have provided the emotional support he needed. This would change the whole dynamic of the Naruto universe as we know it, and there would be no need for him to seek power in dark ways.
One possible plot could be that Naruto's parents being mute adds a new layer of secrecy. They might have a unique way of communicating with Naruto through signs or special ninjutsu. For example, they could use a form of chakra - based sign language that only the Uzumaki family knows. This would make Naruto's relationship with them different from the canon, as he has to learn this new form of communication. And with them alive, they could be protecting Naruto from the shadows, secretly thwarting the Akatsuki's plans against him without his knowledge at first.
In most fanfics, the status of a character's parents can be quite variable. For this 'Harty' in the Hermione fanfic, without having read it, it's difficult to say. It could be that the author decides to keep them alive to have a support system for Harty or perhaps to create a contrast with Hermione's own family situation. If the fanfic is about Harty's adventures in the wizarding world similar to Hermione, having living parents could mean they are either muggles who need to be protected from the wizarding threats or they could be wizards themselves and be part of Harty's journey.
One possible plotline could be that Minato and Kushina raise Naruto as a normal child in Konoha. Naruto would grow up with their love and guidance, which would significantly change his personality. He might be more confident from the start and not have that intense need to prove himself. Also, Minato could train Naruto personally, and Naruto might inherit his father's skills much earlier, leading to different adventures and battles in the ninja world.