For years, the extraordinary ability of parrots to mimic sounds has raised an intriguing but largely unanswered question: what role does this ability play in the wild? This question has been difficult or impossible to study, because most parrots live in dense vegetation and move around a lot, so that no one has been able to get good records of who chirped what to whom, when and under which conditions.
Some recent work makes a start on this problem, as explained by Virginia Morrell, "Why Do Parrots Talk? Venezuelan Site Offers Clues", Science (News Focus) 7/22/2011:
Back in July 1985, while studying raptors on this working cattle ranch [in Venezuela], Steve Beissinger noticed a pair of green-rumped parrotlets nesting in a hollowed-out fence post. “Parrots usually nest high in the treetops, which makes them extremely difficult to study, but these were nesting just a few feet off the ground,” recalls Beissinger, an ecologist at the University of California, Berkeley. “I immediately wondered, ‘Would they nest in artificial boxes?’” […]
Today, the ranch's fence lines look like a parrotlet condo development. Pairs are busy nesting, with females sequestered inside the boxes with eggs and chicks. They emerge when their mates return from foraging with crops full of seeds to share with the family, while all parties exchange short peeping calls. Meanwhile, fledglings congregate in loud, raucous groups in the nearby mango trees, also calling for their next meals. […]
Thanks to Beissinger's nesting boxes, which now number 106, researchers have discovered details of the parrotlets' ecology and life histories, and the project has now entered a new phase focusing on their communicative skills.
For some reason, Science News Focus doesn't give references, but a bit of poking around determines that the work in question was published as Karl S. Berg, Soraya Delgado, Kathryn Cortopassi, Steven Deissinger and Jack Bradbury, "Vertical transmission of learned signatures in a wild parrot", Proceedings of the Royal Society B, Published online before print 7/13/2011:
Learned birdsong is a widely used animal model for understanding the acquisition of human speech. Male songbirds often learn songs from adult males during sensitive periods early in life, and sing to attract mates and defend territories. In presumably all of the 350+ parrot species, individuals of both sexes commonly learn vocal signals throughout life to satisfy a wide variety of social functions. Despite intriguing parallels with humans, there have been no experimental studies demonstrating learned vocal production in wild parrots. We studied contact call learning in video-rigged nests of a well-known marked population of green-rumped parrotlets (Forpus passerinus) in Venezuela. Both sexes of naive nestlings developed individually unique contact calls in the nest, and we demonstrate experimentally that signature attributes are learned from both primary care-givers. This represents the first experimental evidence for the mechanisms underlying the transmission of a socially acquired trait in a wild parrot population.
They started with the background that spectacled parrotlets are believed to have individual "signature contact calls" and to use copies of copies of these signature calls in labelling or naming conspecifics. Thus Ralf Wanker, Yasuko Sugiyama, and Sabine Prinage, "Vocal labeling of family members in spectacled parrotlets, Forpus conspicillatus", Anim. Behav. 2005:
Although there is increasing evidence that signalling animals can refer to objects external to themselves, only weak evidence exists that nonhuman animals use referential signals for different social companions. We tested whether spectacled parrotlets use different acoustic signals for different family members. We recorded two parrotlets interacting with one another during spatial but not visual separation. Discriminant function analysis of the acoustic cues of calls revealed high similarities between calls when both the individual and the interacting partner were loaded together as grouping variables. In playback experiments, the parrotlets were tested with contact calls of a family member recorded during interaction with the tested bird and with calls of the same stimulus bird recorded during interaction with another family member. The birds responded more often to calls uttered in their presence than to calls uttered in the presence of another family member. This suggests that spectacled parrotlets use contact calls to refer to a social companion and thus are labelling or naming their conspecifics. Spectacled parrotlets may thus have mental representations of their social companions, an important ability to live within their complex social system.
Berg et al. note that
If true, this would be an intriguing parallel with humans, in which vocal development is often contemporaneous with parents naming infants. However, one alternative that might appear to be vocal labelling would occur if juveniles acquired their own signature calls independently of parents and siblings, and family members later emulated these calls to solicit attention of a focal individual. Another alternative that might also give the appearance of vocal labelling is if parents provide a variety of signature templates to offspring and, as a result, facilitate individual acquisition of a particular signature call, but without directed labelling.
To begin testing these alternatives, they cross-fostered green-rumped parrotlet chicks and studied the development of their contact calls:
Twelve nests were selected for cross-fostering; pedigrees were used to assure that reciprocating adults were not closely related. Three of these nests failed before any young fledged and were excluded from analysis. Parrotlets in this population have low extra-pair paternity (less than 5%)  and we assumed that on average fathers were significantly more genetically related to individuals in their nest than to nestlings at other nests. Donor/recipient nests were on average 304 m from each other and out of earshot (±s.d. = 189 m, range = 166 m), and had similar first egg dates (average difference = 1 day) and clutch sizes (average difference = 1 egg). […]
They "made audio and video recordings of the nine experimental and eight control nests throughout nestling development", and "similarities between adults and nestlings were quantified with spectrographic cross-correlation and principal coordintates [sic].
Some nestlings in 2007 were studied as breeding adults in 2008, and thus data collected across years were not strictly independent, so we analysed spectrograms for each year separately. We cross-correlated all pair-wise combinations of calls within each year, resulting in two matrices containing 3.0 million and 3.2 million unique cross-correlation values for 2007 and 2008, respectively. […]
They base their conclusions on a fairly complex statistical analysis:
We used PCO analysis on the correlation matrices to extract a series of orthogonal latent variables of the calls (the PCOs). These latent variables described the major axes of variation and were often correlated with traditional spectrographic measurements (e.g. dominant frequency). However, the PCOs have the added advantage of systematically quantifying similarities in the most important dimensions of variability, and their orthogonal nature satisfies independence assumptions in statistical hypothesis testing. […] We used discriminant canonical analysis to determine the combinations of the most salient PCOs for distinguishing calls among social groups (i.e. adults and either biological offspring or cross-fostered nestlings). This method outputs a similarity score for each call (the canonicals). Mean canonical scores were then calculated for each adult, and mates were compared with least-squares regression, which was also used to compare means of nestlings with randomly selected siblings. Mean canonical scores for each breeding pair of adults were paired with the mean of the group of nestlings that they raised.
The results were also complex, though they generally support the notion that nestlings learned vocalizations from their (real or foster) parents:
Calls of adults were more similar to the nestlings they raised than to nestlings at other nests, while controlling statistically for cross-fostering. The interaction of control and cross-fostered groups was weak and non-significant, suggesting that the similarities were not significantly affected by the parents raising cross-fostered nestlings. Adult male calls were more similar to the calls of female nestlings in their nest than to the calls of female nestlings at other nests, but the relationship was only significant in 2007. Adult female calls were more similar to females in their nest than to female nestlings at other nests, but the relationship was only significant in 2007.
This is an encouraging first step. It remains unclear exactly how the nestlings develop their signature calls. Are they random variations on the calls they hear, which the parents later learn to imitate and (perhaps) to associate with individual offspring? Or do the parents play some more active role in assigning "names" to individual nestlings? Is there some sort of back-and-forth process in which the parents encourage the development of nestling-specific call variations, for instance by imitating nestlings' "mistakes"?
And what happens later in life? These parrotlets (like most parrots) form monogamous breeding pairs, who learn or adapt the signature calls of their mates, but no one seems to know much about how that happens, or what else bonded pairs might be communicating to one another. As Morrell's article puts it:
Parrotlets, like other parrots, use contact calls in a variety of ways to communicate with other members of their flocks. For instance, when a male returns to his nesting box, he emits his peep. In response, his mate may make her call, or both their calls, as if saying, “I hear you, Joe. It's Betty here.” Berg suspects that other information is also being communicated. “There is a lot more in that call than ‘Hi, honey, I'm home,’” he says.