Well, they would be very protective parents. Given all the dangerous experiences they've had, they wouldn't want their children to face the same perils easily. At the same time, they'd also be very hands - on in teaching their kids self - defense skills, whether it's combat techniques Percy knows or the architectural knowledge Annabeth has, all to help their kids be well - equipped for any situation.
I think their parenting style would be adventurous. Percy has always been into taking risks and exploring, and Annabeth is smart and strategic. They'd probably encourage their kids to be brave and use their wits, like taking them on quests that are suitable for their age, teaching them about the world of Greek mythology in a fun way.
One possibility is that there is a bit of a culture clash. Percy comes from a world where he's used to the demigod lifestyle and Annabeth has her own set of family values from her mortal family. So when they meet the parents, there could be misunderstandings. Maybe Percy doesn't understand some of the traditional mortal family things that Annabeth's parents expect, like formal dinners or certain etiquettes. And Annabeth might have to explain Percy's rather unorthodox upbringing to her parents, which could be quite an adventure in the fanfiction world.
For a boy, they could go with Orion, a name associated with a great hunter in Greek mythology. It would symbolize strength and skill. And for a girl, they might name her Calypso, not just because it's a beautiful name, but also as a way to remember the different characters they've encountered on their adventures and the lessons learned from them.
One possibility is that Percy arrives at Annabeth's parents' place and is immediately awed by their grand home. He shakes hands with her father firmly and looks her mother in the eye. They might ask him about his adventures, and he'd share some of the less dangerous ones to make them feel at ease. Then they could all sit down for a meal, and Percy would try his best to use proper table manners, all while Annabeth is giving him encouraging looks.
Well, Harry grew up without his parents and in a rather harsh environment at the Dursleys'. He might be extremely protective and affectionate towards his children, making sure they always feel loved. Hermione, on the other hand, grew up in a Muggle family but was always very studious and rule - abiding. She'd probably instill a great love for knowledge in their kids and also strict discipline when it comes to schoolwork. Their parenting style would likely combine Harry's warmth and Hermione's academic focus.
Well, one possible reason could be that Percy made a really stupid decision without consulting Annabeth first. Like, if he went off on a dangerous mission on his own when they were supposed to be a team. Annabeth is a very strategic person and she might feel betrayed or extremely frustrated by such actions.
It could also be due to some sort of magic or mind - control. A powerful enemy might have cast a spell on Annabeth, forcing her to turn against Percy. In this case, it's not really her own will but the result of external manipulation.
In the fanfic, Percy might be really nervous when he meets Annabeth's parents. He could be worried about making a good impression. Maybe he tries to be on his best behavior, like being extra polite and respectful.
One possible reason could be a misunderstanding. Maybe she saw Percy doing something that she misinterpreted as a betrayal of their friendship or their cause. For example, if Percy was seen talking secretly with an enemy, Annabeth might jump to conclusions without knowing the full story.
One reason could be that she's scared. The idea of becoming a mother at that point in her life might be overwhelming, and she might think she can't handle it with Percy. Maybe she wants to protect Percy from the difficulties that come with the pregnancy as well.
One reason could be that Percy often gets into dangerous situations alone to protect others. Annabeth, being his friend or more, would feel bad seeing him constantly at risk.