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 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.
Percy tries to be on his best behavior. He listens carefully when her parents talk, and answers their questions honestly. He also makes sure to compliment their home or any work they've done. He might even offer to help with something small, like setting the table if they're having a meal together. This way, he shows that he's not only interested in Annabeth but also in being part of her family in a way.
In this kind of fanfiction, they might have an adventure together. Maybe they'll face a new monster that they haven't encountered before. For example, a mythical creature from a less - known part of Greek mythology.
Annabeth is known for her intelligence and strategic thinking. She is a strong and independent character. Percy, on the other hand, is brave and has a strong connection to the water due to his father being Poseidon. In the fanfic, when Annabeth sits in Percy's lap, it might be a way to show their unique relationship which has developed over the course of their many shared experiences in the Percy Jackson universe.
Their relationship is likely to be complex. They are friends and comrades who have fought side by side many times. So, there's a strong bond of loyalty between them.
One idea could be that Percy's parents are from different worlds, like his mom is from a modern human world and his dad is from a magical realm. When they meet, there are cultural clashes and misunderstandings. For example, his dad doesn't understand modern technology and his mom is shocked by the magical creatures his dad brings along. Another idea is that Percy has been hiding his true self from his parents, and when they finally meet, he has to come clean about his adventures and powers. This could lead to a range of emotions from disbelief to pride. Or perhaps there's some sort of danger that forces Percy and his parents to meet and work together, and in the process, they really get to know each other for the first time.
Percy might develop more of a steely determination. In Tartarus, he has to face extreme danger, which could make him even more resolute in protecting Annabeth and getting out. Annabeth, on the other hand, could become more resourceful. She'd have to use her wits in this strange and deadly place.
Annabeth might be initially speechless. She'd look at Percy with a mix of disbelief and love. After a moment, she would throw her arms around him and say 'yes' while laughing. She would then start thinking about all the adventures they could have as a married couple and how they could build a future together in both the mortal and demigod worlds.