One possible interaction is that Bardock could spar with Goku and Gohan. He'd test their limits and see how much they've learned about fighting. Another could be Bardock sharing stories of Planet Vegeta with them, which would be really interesting for Goku and Gohan to hear.
A fanfiction might explore the idea of Bardock seeing Goku's life through a time - space rift. He witnesses all of Goku's battles and the friends he makes. Bardock is filled with pride but also realizes that Goku has a different path than the traditional Saiyan way. This makes him question the Saiyan culture he once knew. Meanwhile, Goku starts to have strange dreams about his father, which leads him to search for more about his Saiyan heritage.
A fanfic could explore their relationship through flashbacks. For example, Bardock could have left some sort of message or power for Goku that he discovers later. This would create an emotional connection between them even though they never really met in the original story.
One possible plot could be Bardock's adventures before Goku was born. Maybe it could show how he dealt with the Saiyan politics and his growing concerns about Frieza's rule. This could set the stage for Goku's future battles as his son.
In this fanfiction, Goku might finally learn more about his Saiyan heritage from his parents. He could have heart - warming family moments with Gine and Bardock that were never possible in the original story.
In a 'Dbz Goku meets Bardock' fanfiction, it could start with Bardock somehow being brought back to life or time - traveling to Goku's era. They might initially have a bit of a clash because of their different upbringings. Goku was raised on Earth with love and kindness, while Bardock grew up in the harsh Saiyan society. But as they get to know each other, they find common ground. Bardock could be in awe of Goku's Super Saiyan transformations and Goku could be eager to learn about his Saiyan roots from his father. They could also form a strong bond and work together to protect the universe from new threats.
One possible plot could be that Goku, having achieved new levels of power, somehow travels back in time to meet Bardock. Bardock, initially shocked, quickly realizes this is his son. They could have a sparring session where Bardock sees the full extent of Goku's power and is proud. Then they might sit and talk about Goku's life on Earth, and Bardock could share some of his own past battles and the reasons he sent Goku away.
For Goku, he could gain a deeper understanding of his heritage. Meeting Bardock would make him more aware of his Saiyan side. He might become more confident in his Saiyan abilities. For Bardock, seeing Goku all grown up and powerful could give him a sense of peace. He could realize that his sacrifice was worth it and be proud of the man his son has become.
One potential plot twist could be that Gine reveals that Goku has a special power that was hidden even from the Saiyan elites. This could completely change how Goku views himself and his capabilities.