I think it's important that all parents allow their children to watch Adventure Time.
Visually it's stunning. The crude simplicity and bright primary colors belie a baroque decadence that frequently borders on the psychedelic. There's enough going on in any given episode to singlehandedly cultivate an appreciation for the visual arts, a trait demanded of any well rounded individual.
The subject matter is indeed dramatically dark. This is a story about a boy battling monsters in a post apocalyptic wasteland. Yet this is the only world he knows, and as such he remains blissfully--or as the show says "heroically"--ignorant of the horrors which surround him. What better way to teach children about overcoming their fears? Though his heroism and bravery are lauded by the other characters, to Finn himself there is no heroism because there is no fear--there is only the truth of what must be done. As H.P. Lovecraft remarked, the root of fear is the unknown. By substituting the unknown (where he comes from, what atrocities must have birthed his world, the threat of death and its repercussions) for what is known (that he is a hero and must act heroically) he is able to conquer any fear, a lesson we would all do well to learn.
This complexity of his lack of understanding is used to great effect in each story, resulting in piercingly insightful lessons in morality. Often his heroism is not enough and Finn must deal with realistically unsatisfying choices. Take the Ice King for example. He's not a true villain, simply a lonely old man with an unhealthy way of expressing himself. And time and again as Finn "defeats" him, he's left wondering whether or not he is truly behaving rightly--what makes his "right" better than the Ice King's? Isn't the really "right" thing to do to help the Ice King, rather than beat him up? Such moral ambiguity is further contrasted by the traditional D&D 9 pointed alignment system which the characters all seem to take for granted and make reference to, and Finn's aforementioned simple "heroic" world view. Like Grimm's fairy tales, there really are no easy answers, there are just these stories and how the characters act in them. The viewer is left to decide for themselves what they think. The abrupt and often seemingly non sequitur endings to each episode demand this all the more. Life has no resolutions, there is only the experiences you have and the time you spend reflecting on them, and the candy coating of a make-believe story (whether fairy tale or cartoon) makes that pill a little easier to swallow, preparing a child for the rest of their life.
If I may answer some of the comparisons made: where Ren & Stimpy was certainly raunchy (bordering on pornographic at times) the darkness of that show was almost fully occluded by its absurdity. Hilarious though it was, it made no real demands of its audience the way Adventure Time does. Invader Zim, far darker than either show, opted for much more of a direct social commentary, calling out specific establishments and cultural trends, as compared to Adventure Time's more generally philosophical subject matter. The first generation of "Cartoon Cartoons" (Johnny Bravo, Cow & Chicken, Ed, Edd, & Eddy et al.) each had their own unique issues they grappled with at the outset, but after a season or two (usually coinciding with the departure of the series' creator) watered down to the same level as Spongebob or any other children's cartoon; absurd for absurdity's sake, with no real redeeming cultural value. Adventure Time, on the other hand, has only INCREASED in complexity as the protagonist (and likely the audience) has aged, expanding his awareness, drives, and experience, and beginning to encounter for the first time the drama of interpersonal relationships.
TL;DR: Adventure Time is probably the most important show on television right now, and EVERYONE should watch it, regardless of age.
