A fanfic could revolve around a Hufflepuff whose parents are alive and own a magical business. The Hufflepuff student has to balance school life with helping their parents out during holidays. This could lead to all sorts of adventures as they meet different magical creatures and characters through the business. At the same time, they use the Hufflepuff traits of kindness and patience to deal with difficult customers and situations.
There is a great fanfic that focuses on a Hufflepuff whose parents are alive and are part of the Hogwarts staff. The student has to deal with the pressure of living up to their parents' reputation. The story is full of the Hufflepuff's journey to find their own identity within the school while still respecting and learning from their parents. It also has some really interesting sub - plots about the Hufflepuff's relationships with other houses.
A fanfic with Harry's parents alive could have a plot where they discover a new threat to the wizarding world. Since they survived Voldemort's first attack, they are more vigilant. They raise Harry to be a strong wizard, teaching him both powerful spells and the values of the wizarding world. The story could involve them going on adventures together to find the source of this new threat. Along the way, Harry learns more about his family's history and his parents' experiences at Hogwarts. There would be a lot of family bonding and magical discoveries.
One great fanfic is 'The Potter Family Reunion'. It imagines how different Harry's life would be with his parents around. His parents, James and Lily, are actively involved in his upbringing, and he gets to grow up in a loving family environment. It also delves into how the wizarding world would change with the Potters alive, as they were important figures. There are interesting interactions between Harry and his parents, and it shows how they would have trained him in magic from a young age.
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.
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.
A typical plot element could be the conflict that arises from their presence. Some in the wizarding world might be suspicious or even afraid of the idea of the Potters being brought back to life. There could be a struggle between those who support Harry and his parents and those who think it's against the natural order of things. This can create a lot of drama and tension in the fanfic, making it an exciting read.
You can try some popular fanfiction websites like Fanfiction.net or Archive of Our Own (AO3). Just search for 'Harry summons parents alive' in their search bars, and you should be able to find relevant stories.
It could be that the Order has a plan that only Harry can be a part of. His unique experiences and skills, like his ability to resist the Imperius Curse and his connection to Voldemort, are valuable. So they summon him to put these to use in their fight against the Dark side.
A popular one is 'A World with the Potters'. In this fanfiction, with his parents alive, Harry has a happy and normal childhood at home. He doesn't have to endure the hardships at the Dursleys. His parents are actively involved in the fight against Voldemort and they train Harry to be a powerful wizard together.