Gas-rich galaxies in dense environments such as galaxy clusters and massive groups are affected by a number of possible types of interactions with the cluster environment, which make their evolution radically different than that of field galaxies. The dIrr galaxy NGC 1427A, presently infalling towards the core of the Fornax galaxy cluster, offers a unique opportunity to study those processes in a level of detail not possible to achieve for galaxies at higher redshifts. Using HST/ACS and auxiliary VLT/FORS ground-based observations, we study the properties of the most recent episodes of star formation in this gas-rich galaxy, the only one of its type near the core of the Fornax cluster. We study the structural and photometric properties of young star cluster complexes in NGC 1427A, identifying 12 bright such complexes with exceptionally blue colors. The comparison of our broadband near-UV/optical photometry with simple stellar population models yields ages below ~4x10^6 yr and stellar masses from a few thousand up to ~3x10^4 Msun, slightly dependent on the assumption of cluster metallicity and IMF. Their grouping is consistent with hierarchical and fractal star cluster formation. We use deep Ha imaging data to determine the current Star Formation Rate (SFR) in NGC 1427A and estimate the ratio, Gamma, of star formation occurring in these star cluster complexes to that in the entire galaxy. We find Gamma to be among the largest such values available in the literature, consistent with starburst galaxies. Thus a large fraction of the current star formation in NGC 1427A is occurring in star clusters, with the peculiar spatial arrangement of such complexes strongly hinting at the possibility that the starburst is being triggered by the passage of the galaxy through the cluster environment. ; Comment: 14 pages, 15 figures, accepted for publication in The Astronomical Journal. Updated document