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.
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.
To create engaging characters, start with giving Percy's parents distinct personalities. For his mom, you could make her really strong - willed and intelligent, maybe a bit overprotective. His dad could be more laid - back but with a hidden depth of power. Also, give them quirks. For example, his mom always has a lucky charm she fiddles with when she's nervous, and his dad has a unique laugh. Their appearance can also play a role. If his mom has bright red hair and his dad has a long, flowing beard, it helps the readers visualize them better.
One possible plot is that Percy meets Paul's parents during a family gathering. He feels out of place among all of Paul's relatives. However, he tries his best to be polite and helpful. He helps in the kitchen or with setting up things. Through these small acts, Paul's parents begin to see his good nature. As the gathering progresses, Percy tells some funny stories about Paul that make everyone laugh and he becomes more accepted.
In the fanfic, Percy's journey to meet his parents is filled with anticipation. He has changed so much since he last saw them. As he enters their home, there's a sense of familiarity yet also a bit of strangeness. His parents look at him with a mix of surprise and joy. Percy shares his tales of heroism, but also his fears and insecurities. His mother wipes away a tear, and his father pats his back, assuring him that they are always there for him, no matter what.
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.
I haven't actually read this specific fanfic. But generally, if 'camp camp' meets Max's shitty parents in a fanfic, it could be a really interesting exploration of the characters. It might show how Max's relationship with his parents affects his experiences at camp, and how the camp environment in turn influences his view of his family situation.
Well, it could be that Harry meets his parents through some magical time - turner accident. Maybe he's on a mission in the past and stumbles upon them.