Hank "Rusty" Truman asked: what are their parenting styles? Who is the more strict one?
Azula is the strict one 90% of the time. She doesn't expect them to be perfect (she knows better than anyone how unfair that can be, and how it screws with a kid's mind even if they do manage to be "perfect") but she does expect them to be hard-working, persistent, responsible and to do their best. She will help them if they're struggling, but she doesn't let the fact that they're her babies affect her judgment. The kids find it very amusing that she treats Zuko the exact same way.
The times she actually relaxes a bit, are when Zuko has to step in. Their kids are good, kind children... but they're also just as proud, stubborn and temperamental as their parents, and can go too far when they want something or when they feel they were wronged. Azula has become a much better person, she's still Azula and wants them to have anything they want and praises them for standing up for themselves... no matter how they did it, meaning Zuko needs to both teach his kids morals and remind his sister-wife that that very concept exists (because I love the idea the Azula is forever the shoulder devil/blue dragon corrupting him, while also being saved by him).
They're both very proud of everything their kids accomplish - that goes from "Look! Our baby girl said her first word!" to "Look! Our baby girl is Fire Lord!" and they make sure to demonstrate as much as possible because they don't want their kids to ever feel the way their parents made them feel. They also make sure that any rivalry between their kids doesn't go beyond regular sibling rivalry.
Their kids will have the best teachers/masters ever, will learn how to defend themselves and each other, will have their full support to pursue any interest they have, and anyone that wants to be friends with or date their kids will have to prove that they truly want the best for them, and if Zuko and Azula find out their babies were hurt or betrayed in any way, heads will roll.
Pretty much the only things their kids can complain about is that:
1 - They can be a bit overprotective.
2 - They're super affectionate to each other and do not give a damn that their kids are still in the room.