There's a debate to be had with regards to Dragon Ball Z's Goku as a father to Gohan. He spent much of the boy's early years either dead, off-world, training or somewhere else he couldn't be a father. Despite this, however, Gohan, knowing that his father still loves him, is able to grow up healthily both physically and mentally. Other anime protagonists are similarly unaffected by parental absence, which is part of why it's so hard to say why they're removed in the first place.
Many anime characters go through their series having one or both of their parents missing from their lives. This can happen for all sorts of reasons and thus has all kinds of effects on the characters. Understanding why this is such a common trend in anime will require a closer look at how or why it happens and what impact it has on the story.
Perhaps the most common way for anime characters to lose their parents is for them to die tragically. If it happens when the protagonist is too young to remember, then it won't really mean anything to them. If they do remember, it's generally a traumatic experience that twists them on the inside. This key context behind parental death can greatly change how the characters think and feel about it.
A good example of this contrast would be with Naruto's titular character and his friend Sasuke Uchiha. For Sasuke, having his parents and the rest of his clan killed caused him such great pain that he spent the entire series seeking revenge. Naruto, on the other hand, never got to know his parents, so he wasn't really influenced by their deaths, nor could he understand the pain of losing them. Sasuke called him out on both of these points during their first fight at the Final Valley. The difference between when and how one grows up without parents can create a broad spectrum of reactions.
If a protagonist loses their parent(s) because they were killed, then it can become the driving force behind their actions. This often manifests as a strong desire to defeat the culprit, typically for revenge. For those who recognize Sasuke as a deuteragonist, there are plenty of protagonists who share this distinction; popular examples include Eren Yeager (Attack on Titan), Lelouch Vi Britannia (Code Geass), Shinn Asuka (Gundam SEED Destiny) and Rin Okumura (Blue Exorcist). For these protagonists, losing their parents and parent figures is a source of both great pain and great anger that they can hopefully overcome by their series' ends.
Another more lighthearted cause of absence comes from the harem protagonist’s parents. In harem anime, parents are kept out of the house so that it’s easier for the protagonist and their girlfriends to get into all sorts of ecchi hijinks. They could be busy with work or jet set for most of the series. Such series include Date A Live and Monster Musume.
Unfortunately, parents being away from home because they're busy with work is a common phenomenon not just in anime but in Japan as a whole. The best representation of this in anime comes from Neon Genesis Evangelion. In this series, Shinji Ikari's father Gendo keeps his distance because of how important his job is, but also because he genuinely doesn't know how to communicate with his son. This form of neglect leaves Shinji with horrible anxiety and a lack of self-worth.
Another type of absent parent is the deadbeat. They're alive but they don't raise their children for one reason or another. Goku arguably qualifies as this, but a better example would be Hunter x Hunter's Ging Freecs. He left his son Gon and the rest of his family so he could pursue his goals as a Hunter. Rather than take issue with his abandonment, Gon became curious about Hunters and became one with the hopes of finding his father. Whether the resulting quest is good or bad is debatable, but Gon seems happy with his decision.
A more positive look at missing parents is how protagonists compensate for them. Even without parents, they can still lead fulfilling lives with good friends and other people who care about them. This holds true for Naruto and many other shonen protagonists; Monkey D. Luffy (One Piece), Natsu Dragneel (Fairy Tail) and Goku. In this sense, these characters are almost better off for not having parents.
However it happens, there are plenty of things that can happen if an anime protagonist doesn't have their parents. It all depends on what story needs to be told and how absentee parents are necessary for moving the plot forward. It usually has a negative effect on the protagonist, but not always. The responses to these absences are just as varied as the people who experience them.