Three Little Kami from School

Question: Are Amaterasu, Susanoo, and Tsukuyomi considered the children of Izanami, or is Izanagi their one and only parent?

That depends entirely on which version of the myth you prefer, because both are legit.

In the Kojiki, one of the primary sources for Japanese mythology, the three are considered the children of Izanagi alone; after his escape from Yomi, pursued by his grieving and terrible wife, he stopped to purify himself of the uncleanliness of the Underworld by washing in a stream. When he washed his face, Susanoo was born from the washing of his nose, Tsukuyomi from the washing of his right eye and Amaterasu from the washing of his left eye. They're the symbolic result of his act of cleansing himself; purifying himself is such a strong and resonant act that it causes the three most powerful deities of the pantheon to be born.

In the Nihon Shoki or Nihongi, however, one of the other primary sources for Japanese mythology, Izanagi and Izanami decide together before her death that they should create the most splendid of kami to rule the heavens, and together give birth to Amaterasu and Tsukuyomi, one each to rule the day and night. In this version, Susanoo is not directly mentioned, but he's considered Amaterasu's brother, so it would be pretty safe to assume that he was also one of the natural-born children of the first divine couple.

And, to make matters even more confusing, the Nihongi also contains another version of the birth of the major gods, in which it is Izanagi alone who creates Amaterasu and Tsukuyomi, but he does so by creating them from the reflections of two mirrors he holds in either hand. Susanoo goes unmentioned again; he may be one of the Three August Siblings, but he's also clearly not great rulership material, which is probably why he doesn't always turn up in accounts of how the divine rulers came to be appointed.

(Incidentally, it's also the Nihongi that suggests that Amaterasu and Tsukiyomi were created specifically for one another and are or were lovers and consorts as well as siblings. Ooh la la! Work that into your story!)

So the exact parentage of the three premier Japanese gods is up in the air somewhat, and really depends on what version of the myth you prefer. Amaterasu and Tsukuyomi are always born together, with Susanoo only sometimes included (but still their sibling). If you prefer the idea of them as sovereign rulers of the heavens, their parentage by Izanagi and Izanami is probably the version you want, whereas if you prefer the idea of them fundamentally opposed to Izanami and everything she stands for, you may want to instead use the version where they have nothing to do with her, even pre-death.

A particularly interesting setup might be to consider Amaterasu and Tsukuyomi Izanagi's parthenogenic children, but Susanoo a child of both parents, which would explain why he isn't mentioned in that myth and why he so strongly decries his father for leaving Izanami trapped in the Underworld. He would still be the other twos' sibling through their shared father, but the genetic divide would explain why he never gets along with them (and him being Izanami's son while the other two weren't would also explain his death connotations, which they decidedly don't share!).

Personally, I'm a fan of the idea that they're all the children of both ancient parents, but there are plenty of ways to go with it. The Kojiki myth of all three being created by Izanagi is the most well-known, but the alternate versions from the Nihongi are just as legitimate (and might be older, considering that the Nihongi preserves some older folklore as opposed to the Kojiki, which aimed to avoid extra versions in the name of creating an "official" document for the entire country).