And also the absence of motherhood in both series.
On the father-child relationships, we have Ozai and Yakon as examples of Really Evil Parents: both used their children essentially as tools to cement their own power. Both played their children off one another (though Yakon might not have done so as calculatingly as Ozai). The outcomes were different -- in Yakon's case, his sons were far more loyal to each other, while in Ozai's, his children competed for his favor until Zuko left -- but the thing they had in common was that all of their children were marked by their fathers, and that three of the four eventually turned against their fathers.
On the opposite end we have Hakoda and Tenzin as good parents, despite circumstances in the case of Hakoda. Gyatso and Iroh (and Tenzin again) might fall in here since all of them acted as father-figures to the cast. (Heck, Iroh's presence might be why Zuko had enough awareness to figure out that he didn't need Ozai's fatherly approval, since he had the approval of a man far better at fathering than Ozai would ever be.)
In the middle, you have Lai Beifong as 'good but misguided' and Hiroshi Sato as 'once good, but changed by grief'. (Azulon was not onscreen enough to really define where her falls, but by definition, he's probably not in the Good Father category, since he did order one of his grandkids killed because his son was being an ass to his other son; leaning towards Really Evil Parent.) Which makes things a bit more complicated; then again, so do the Really Evil Dads, since, well... it's your dad.
(Interesting that we have a few good, one bad and one ambiguous in both Avatar and Korra. I would have loved to see Zuko and Tarlok sit down and chat about their family issues.)
Brotherhood... or siblinghood also seems to show up. We have Mako and Bolin; Sokka and Katara; and the three airbender kids (especially Ikki and Jinora) as heroic siblings, and Tarlok and Noatuk/Amon and Zuko and Azula as antagonists that are both against the heroes and playing off each other (though in the former case, without one half being aware of it). We also have Ozai and Iroh forming the two sides of the Fire Nation (again, as antagonists, even if Iroh was only one because he followed Zuko around in Season 1), and the hints of Tenzin and Bumi the Younger's brotherly relationship as players of the between generation.
But, mother-child relationships seem to be less prominent, or, when prominent, are because of Dead Mother syndrome. Zuko (and maybe Azuka), Katara, Sokka, Mako, Bolin, and Asami are all strongly affected by an absent mother (in Mako and Bolin's case, both parents). Toph's parents are defined as a unit most of the time. Tarlok and Amon's mother doesn't get much attention, besides that Tarlok (like Zuko) had a strong bond with her. Korra clearly loves her parents, but they don't get screentime. While we see Toph, and Lin, we have little indication of what their relationship was like. Katara and Pemma are the mothers who get the most screentime in Korra, and they still seem overshadowed by Yakon and Tenzin.
(Gran-gran Kanna also has a maternal role to Katara and Sokka, but is not prominent in Avatar.)
We don't nearly have the same kind of dead father syndrome; Mako and Bolin lost both parents, and Katara and Sokka are affected by Hakoda's absence... but he shows up again on camera unlike, say, Ursa. We don't even know who Lin's father is. I can't think of an example of a character with a dead or absent father but a present mother (unless you count that Yakon died before his wife) -- while there are several of the latter.
It's kind of frustrating, because the series does a good job of exploring parental issues, but only from one side. Mothers seem to fall into two categories: the nurturing caregiver or absent. Others (Toph, for sure) just aren't shown.
On the father-child relationships, we have Ozai and Yakon as examples of Really Evil Parents: both used their children essentially as tools to cement their own power. Both played their children off one another (though Yakon might not have done so as calculatingly as Ozai). The outcomes were different -- in Yakon's case, his sons were far more loyal to each other, while in Ozai's, his children competed for his favor until Zuko left -- but the thing they had in common was that all of their children were marked by their fathers, and that three of the four eventually turned against their fathers.
On the opposite end we have Hakoda and Tenzin as good parents, despite circumstances in the case of Hakoda. Gyatso and Iroh (and Tenzin again) might fall in here since all of them acted as father-figures to the cast. (Heck, Iroh's presence might be why Zuko had enough awareness to figure out that he didn't need Ozai's fatherly approval, since he had the approval of a man far better at fathering than Ozai would ever be.)
In the middle, you have Lai Beifong as 'good but misguided' and Hiroshi Sato as 'once good, but changed by grief'. (Azulon was not onscreen enough to really define where her falls, but by definition, he's probably not in the Good Father category, since he did order one of his grandkids killed because his son was being an ass to his other son; leaning towards Really Evil Parent.) Which makes things a bit more complicated; then again, so do the Really Evil Dads, since, well... it's your dad.
(Interesting that we have a few good, one bad and one ambiguous in both Avatar and Korra. I would have loved to see Zuko and Tarlok sit down and chat about their family issues.)
Brotherhood... or siblinghood also seems to show up. We have Mako and Bolin; Sokka and Katara; and the three airbender kids (especially Ikki and Jinora) as heroic siblings, and Tarlok and Noatuk/Amon and Zuko and Azula as antagonists that are both against the heroes and playing off each other (though in the former case, without one half being aware of it). We also have Ozai and Iroh forming the two sides of the Fire Nation (again, as antagonists, even if Iroh was only one because he followed Zuko around in Season 1), and the hints of Tenzin and Bumi the Younger's brotherly relationship as players of the between generation.
But, mother-child relationships seem to be less prominent, or, when prominent, are because of Dead Mother syndrome. Zuko (and maybe Azuka), Katara, Sokka, Mako, Bolin, and Asami are all strongly affected by an absent mother (in Mako and Bolin's case, both parents). Toph's parents are defined as a unit most of the time. Tarlok and Amon's mother doesn't get much attention, besides that Tarlok (like Zuko) had a strong bond with her. Korra clearly loves her parents, but they don't get screentime. While we see Toph, and Lin, we have little indication of what their relationship was like. Katara and Pemma are the mothers who get the most screentime in Korra, and they still seem overshadowed by Yakon and Tenzin.
(Gran-gran Kanna also has a maternal role to Katara and Sokka, but is not prominent in Avatar.)
We don't nearly have the same kind of dead father syndrome; Mako and Bolin lost both parents, and Katara and Sokka are affected by Hakoda's absence... but he shows up again on camera unlike, say, Ursa. We don't even know who Lin's father is. I can't think of an example of a character with a dead or absent father but a present mother (unless you count that Yakon died before his wife) -- while there are several of the latter.
It's kind of frustrating, because the series does a good job of exploring parental issues, but only from one side. Mothers seem to fall into two categories: the nurturing caregiver or absent. Others (Toph, for sure) just aren't shown.