Brood Parasitism - Parasitism/Ammensalism/Commensalism?

The cuckoo chick hatches earlier and instinctively pushes the host eggs out of the nest. If you view this from the genetic perspective of the crow, it becomes the foster parent of offspring that does not carry its genes. The harm, therefore, is to the host's passing on of genes, rather than to the immediate health or well-being of the individual.

For more information see:

• Brood parasite (Wikipedia)
• The Ecology of Avian Brood Parasitism (Nature)

Various aspects of this question suggest that OP is not asking about cuckoo/host interactions in general, but about research published in 2014 which appeared to show carrion crows (Corvus corone corone) in an area of Northern Spain actually benefitting from the presence of a great spotted cuckoo chick (Clamator glandarius) in the nest.

Unlike other species of cuckoo, clamator glandarius chicks do not "evict" the host eggs and chicks by pushing them from the nest. Furthermore, they are actually notably smaller than the crow chicks, making it harder to compete with them for food and host-parental attention.

Based on a 16-year dataset of observations, the researchers found that the average number of crows successfully fledged in a parasitized nest was $1.584$, versus $1.379$ for an unparasitised nest, suggesting an advantage to the cuckoo chick's presence. This particular cuckoo species had other hosts in the region, notably magpies (Pica pica), and no such advantage was observed for magpie hosts.

When attacked, chicks of this species could secrete

a malodorous cloacal secretion ... a mix of caustic and repulsive compounds, dominated by acids, indoles, phenols, and several sulfur containing compounds ... that are known to repel mammals and birds

The researchers conjectured that this ability bestowed a degree of protection on the other chicks in the nest, by repulsing predators and causing them to abort their attacks prematurely. Experiments showed that crows, raptors and mammalian predators were more likely to avoid meat that had been treated with this substance.

Other experiments were carried out, such as introducing cuckoo chicks to unparasitised nests and observing the effect this had on crow chick fledging rates, which appeared to support the aforementioned findings.

Based on this, the researchers suggested that this relationship may be more mutualistic than parasitic. To quote the paper:

We believe that the outcome of this parasite-host interaction may depend on predator pressure and thus fluctuates among parasitism, commensalism, and mutualism. ... It has been advocated that interspecific interactions should not be strictly categorized as parasitism, commensalism, or mutualism (21), because costs and benefits for each partner may vary in space and time, producing variable outcomes depending on the environmental context.

However, these findings have yet to be replicated. In particular, a study was published in 2017 in which another research group studied great spotted cuckoo parasitism of magpie, blackbird and common crow nests in southern Spain. They had four years worth of data for the crows, and eight for the magpies. This group did not observe the same benefits in terms of predator repulsion, or in the average number of fledged crows.

With regards to the secretions, they noted:

Great spotted cuckoo nestlings do not excrete the malodorous secretion when they are scared in the nest (personal observation), as does happen with hoopoe (Upupa epops) nestlings that are capable of throwing their faeces and cloacal secretions directionally against the predator [35], but only when they are grabbed (personal observation).

this secretion did not protect parasitized blackbird nests

direct evidence of a predator releasing a cuckoo chick is also lacking today. During nearly 30 years of studying great spotted cuckoos, we have ... never found any injured cuckoo fledgling in either magpie or crow nests.

the protection of crow nestlings would occur only in the case that predators attack the cuckoo chick first, which presumably would happen only in a fraction of predation attempts.

They also criticised the experimental methodology, claiming that the relative developmental stage of the introduced cuckoo nestlings did not match that which would occur in the wild.

The autors of the original study wrote a response in which they put forward counterarguments to some criticisms (for example, the same size in terms of number of nests being smaller in some years than others). In the case of the introduced-cuckoo-chick experiments, I am not entirely convinced by the counterargument, but am definitely not an expert in this field.

The authors of both studies agreed that further research with more data on different populations was needed.

In the aforementioned response, the authors stated:

Interestingly, a positive effect of the presence of cuckoo chicks on magpie nest success was reported in a different population in southern Spain (Sierra Morena) where carrion crows are rare and are not an important predator of magpie nests

However, the 1998 paper they cited in which this was supposedly observed is not available online (not even behind a paywall!) and so I can't offer any opinion on this. I have, though, looked at some of the other papers that cite it and have not seen any reference to this beneficial effect.

In summary, there may, depending on further research, be evidence that this particular cuckoo/host relationship has commensalistic aspects which are more significant than the parasitic. Or ammensalistic, if the differences in crow chick survival rates are not statistically significant. But this is not guaranteed, and future research with other datasets may show that the relationship is mostly parasitic.

Sources:

Canestrari, D., Bolopo, D., Turlings, T. C., Röder, G., Marcos, J. M., & Baglione, V. (2014). From parasitism to mutualism: unexpected interactions between a cuckoo and its host. Science, 343(6177), 1350-1352.

Soler, M., de Neve, L., Roldán, M., Pérez-Contreras, T., & Soler, J. J. (2017). Great spotted cuckoo nestlings have no antipredatory effect on magpie or carrion crow host nests in southern Spain. PLOS ONE, 12(4), e0173080.

Canestrari, D., Bolopo, D., Turlings, T. C., Röder, G., Marcos, J. M., & Baglione, V. (2017). Formal comment to Soler et al.: Great spotted cuckoo nestlings have no antipredatory effect on magpie or carrion crow host nests in southern Spain. PLOS ONE, 12(9), e0184446.

The Sierra Morena paper I couldn't find online:

Arias-de-Reyna, L. (1998). Coevolution of the great spotted cuckoo and its hosts. In: Rothstein S. I., editor. Parasitic Birds and Their Hosts, Studies in Coevolution. Oxford Ornithology Series, 9, 129-142.