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I'm not quite sure what's going on with that, I can't wait until it's explained since right now it's just a big question mark.

Unless I missed something, we've got the flash to the other character and Gai's "you look so much like her" (or something like that) line from episode 5. Am curious.

Let's see what I can do for you, Rusty.

 Spoiler

So Himari is/was sick. She died, really. Then the magic penguin hat saved her life at the whim of some crazy supernatural person who wants the "Penguin drum". Penguin hat lady orders Shoma and Kanba to find the penguin drum if they want Himari to continue to live. The penguin drum is supposedly Ringo's diary. Ringo has mad hots for Tabuki, a childhood friend of her dead sister, Momoka. Ringo wants to become Momoka because she has some serious inferiority complex at the start of the show. Tabuki also happens to be Shoma and Kanba's homeroom teacher.

Ringo can't really get with Tabuki because Yuri, another childhood friend of Momoka, goes out with then marries Tabuki. Both Tabuki and Yuri were saved from absurdly pomegranatety childhoods by Momoka, so the marriage is kinda for show. It's a kinda crazy bisexual love-triangle thing, and both Tabuki and Yuri end up getting noodly with Momoka to some extent.

So everyone hates Kanba/Shoma/Himari's parents because they lead a terrorist organization, and in one of their attacks they killed Momoka. Momoka is the only one that died because she took it upon herself to save everyone else since she has some fate-changing powers (which she used to save Tabuki and Yuri) that comes at a sort of "eye for an eye" cost to herself.
The father is probably not actually dead, just missing. So Tabuki damn near kills Himari in an attempt to get Kanba to reveal the location of the father.

The pink haired guy, Sanetoshi, hasn't been very well explained but is some sort of fate/god figure like hat lady. After Himari dies a second time, his medicine brings her back, but Kanba has to keep paying tons of money for it. It was speculated earlier in the thread that he's getting the money from his father.

Then there's Masako, who has/had an interest in Kanba. She's also after the penguin drum for her brother Mario who is in a similar circumstance as Himari. There's also speculation she's the half-sister of the main three.

The penguins are a byproduct of the whole penguin hat/drum thing, and seem to exist as flavor and humor for the show. They're generally helpful to the main three.

So Himari is kind of the accidental centerpiece of this ridiculous, tangled mess of a diary that may or may not control fate, Tabuki, Yuri, and Ringo's desires to bring back/avenge Momoka (who was killed by the main three's parents), and Kanba and Sho's desires to save Himari (Kanba more or less has an incestuous interest in Himari). And then there's the pink haired guy, Sanetoshi.

This show is crazy.

Edit: And on this subject
Penguindrum 19:
This show is crazy.

 Spoiler

So the parents are alive. Or Kanba's delusional. But I'm thinking alive. Especially with the "I kept the secret" flashback text. Rather than Masako being a half-sibling, perhaps she was originally the sister and Himari's fate was transferred with hers.