So, there are two parts to this poem and they don't really look anything alike, as far as form is concerned. The first part is a loose villanelle form, the second part is in free verse.
The first bit of the poem – titled "The Cane Fields" – is, in fact, an unrhymed villanelle. (Villanelles are a kind of poem that, traditionally, have a pretty strict ABA / ABA / ABA rhyme scheme (think "cat/ball/bat flat/hall/rat"). So what we're saying here is that Dove is breaking some of the rules. But, hey, there are no poem police here.) For a good explanation of the traditional villanelle form, and how it's been used by more contemporary poets (the form kicked off in the 1800s, so it's pretty old), see our guide to "One Art," by Elizabeth Bishop.
OK, now that you've read that, there's not a whole lot more to say except for the fact that Dove has two refrains – "there is a parrot imitating spring" and "out of the swamp the cane appears" – and that she doesn't bother to rhyme her stanzas. The non-rhyming fact is a curious one. One of the effects that it has on the poem is that the form sort of sneaks up on you. It's subtler. It's like you're reading, and then you suddenly think, "Hey! I've seen this line before," and then you go back looking for the form. It's a quieter gesture than a strictly rhyming villanelle.
Now about the second part, titled "The Palace." Dove has said that the second bit was originally supposed to be a sestina, which is a really wacky form of poetry involving six-line stanzas and lots of rules (source). But since the second part isn't actually a sestina, we won't go into it here. Instead, Dove writes in free verse (no form or rhyme scheme) but repeats some of the lines from the first part, to keep the poem tied together and feeling kind of driven and insistent. Her lines are relatively tightly controlled. They're all about the same length and form a neat rectangle on the page, and they range from about six to thirteen syllables. They don't have a particularly set rhythm – no iambic pentameter here.
So this is half-form, half-not poem, and overall it's pretty loose. We think the important part here about the form is the repetition that we've mentioned. It's a haunting, of sorts – one that Dove is playing around with form to emphasize without being too overbearing about it.