Parenting a young child presents many tough choices, but few are so fraught with compromise, so much a decision between lesser evils, as choosing what a child should listen to (and watch, as most children’s albums are also DVDs). The conventional choices all have their crippling defects: the pre-verbal psychedelics of the Teletubbies and Boohbahs, the unrelenting cheer of the Wiggles, the muzak classics of Baby Mozart. And then, in a whole other category of cloying, there’s Barney. To sit through these albums once will try your patience; to suffer through them the hundreds of times a child demands will test your sanity.

To the relief of parents everywhere, an alternative has emerged in the last few years; one that is to mainstream children’s music what indie rock is to pop, and draws its talent from indie rock’s ranks. Call it indie romp.

The fledgling genre already has its stars: Dan Zanes, Trout Fishing in America, and Ralph’s World, to name a few. But the most successful of the bunch is They Might Be Giants. Best known for quirky, early 1990s hits such as “Birdhouse in Your Soul” and “Istanbul (Not Constantinople)” TMBG began making children’s music in 2002 with the album/DVD “No!” It quickly jumped to no. 1 on the Billboard Children’s Music Chart and has sold some 180,000 copies to date.

Ever since, the band has been leading a fascinating double career as indie rock stalwarts and cradle rock upstarts. But in the music of TMBG, the two are intertwined. The adult music has always been playful and regressive, and the children’s music is surprisingly sophisticated. “Our music is strategically designed not to drive adults crazy,” said John Flansburgh, one half of the band.

This past February, TMBG released its second children’s album and DVD, titled “Here Come the ABCs.” As the name implies, the songs are all alphabet-themed; the lessons, however, are much broader. “Our goal isn’t to educate kids, but to spark kids’ imaginations,” said Mr. Flansburgh. “Kids are going to learn to tie their shoes and learn the alphabet somewhere else. What we have to offer as songwriters is something to make them realize that there’s a whole colorful world of ideas and culture out there waiting for them, that growing up is something to be excited about.”

“Here Come the ABCs” has many of the trappings of other children’s products. The songs are set to whimsical videos with Muppet-like characters; there are hip animations that lead youngsters through a variety of games and sing-a-longs. It’s the style and sensibility that set it apart.

Like the Lemony Snicket books, “Here Come the ABCs” challenges children with sly jokes, clever wordplay, and baroque concepts. “We don’t apply that condescending ‘now I’m singing at you’ thing,” said Mr. Flansburgh, though he added, “It’s more a side effect than a strategy, to be perfectly honest. It’s the only way we know how to write.”

Take the lyrics to “Fake Believe,” for example: “We’re wearing fake fur and riding on alpacas/ On the wild frontier, wearing wax mustaches/ Pretending we’re cowhands, yodeling like cowhands do/ Yo-da-lay-he-hoo.” For a child learning the rudiments of language, this is pretty heady stuff.

But to Mr. Flansburgh’s way of thinking, it’s just the stuff to hold their attention. “Kids interpret the world all the time and they’re used to only understanding a percentage of what’s going on,” he said. “A lot of stuff that’s designed for kids goes so far out of its way to be completely explicable that it’s not at all entertaining for the kids.”

In this, TMBG connects with a much older tradition of intelligent, independent children’s music: folk. Folk singers such as Woody Guthrie, Ella Jenkins, and Pete Seeger wrote and performed traditional songs for children in the 1940s, ’50s, and ’60s. And it’s this tradition that provided the template for TMBG.

Mr. Flansburgh was raised in the shadow of “the very kid-oriented” Cambridge, Mass., folk scene. “Growing up with something like that was such a good example of how to do it, an unneurotic example,” he said. “It came from natural place from them. It was done with love. That’s the X-factor. When people are doing work for hire, people can see through it – kids especially.”

As refreshing as this approach is for parents, it’s equally refreshing for the band. “You only have to be good,” Mr. Flansburgh said. “You don’t need any spin. You don’t need to be fashionable. You don’t need to be particularly good-looking.”

Since they began recording children’s music, the band has enjoyed a creative renaissance. You might say they relearned the very lessons they intend for children to take away. “Our adult and children’s stuff has been the better for it,” Mr. Flansburgh said. “It’s good for us to remember that there’s a larger world of culture and ideas as well.”