The relationship could develop in a more emotional way. Maybe Trunks gets injured badly in a fight, and Vegeta realizes how much he cares for his son. He then becomes more protective and affectionate towards Trunks. On the other hand, Trunks could also start to understand his father better, seeing that Vegeta's tough love is actually a way of showing his love and concern. This new understanding could lead to a more harmonious and loving relationship between them.
It could be a genetic mutation in the fanfic's storyline. Some strange event or factor could have caused a change in Bardock's genes, making it possible for him to become Super Saiyan. This genetic change might be related to some ancient Saiyan lore or a new element introduced in the fanfic's universe.
One possible analysis could be that it might explore the complex relationship between Vegeta and Trunks. Maybe in this fanfiction, Vegeta's strict training methods could be seen as 'hurting' Trunks in a sense, like pushing him too hard physically or emotionally. It could also delve into how Trunks copes with that and if it ultimately makes him stronger.
Well, if Naruto goes Super Saiyan 3 in a fanfiction, he would likely gain an insane amount of power. His physical abilities would skyrocket. He could move at incredible speeds, and his strength might be enough to shatter mountains. His chakra would probably combine with the Saiyan power in some unique way, creating new and powerful jutsu.
Well, in this fanfiction concept, Naruto going Super Saiyan 4 could also lead to some interesting character dynamics. His friends might be initially shocked and a bit scared of this new form. Naruto would then have to prove that he's still the same person despite this radical transformation. And as for the enemies, they would surely be caught off - guard and have to quickly come up with new strategies to deal with this super - powered Naruto.
The change of Bardock becoming Super Saiyan in the Vegeta fanfic can completely rewrite Saiyan history. It could start a new era of Saiyan pride and power. Bardock, with his new - found Super Saiyan form, might be able to teach the younger Saiyans about the transformation. This could lead to a more united Saiyan race, as they all work towards achieving the Super Saiyan state. It might also mean that they are not as easily defeated by Frieza, as they have a stronger force to oppose him. This could result in a different galactic power structure, with the Saiyans being a more dominant force.
Vegeta would also be an important character. His rivalry with Goku would intensify even more. Seeing Goku achieve the Super Saiyan form early would fuel his jealousy and his desire to become stronger. He might even try to find new ways to train or gain power to catch up with Goku.