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Context. Many questions on the nature of astrophysical counterparts of high-energy neutrinos remain unanswered. There is increasing evidence of a connection between blazar jets and neutrino events, with the flare of the gamma-ray blazar TXS0506+056 in spatial and temporal proximity of IC170922A representing one of the most outstanding associations of high-energy neutrinos with astrophysical sources reported so far. Aims. With the purpose of characterising potential blazar counterparts to high-energy neutrinos, we analysed the parsec-scale regions of gamma-ray blazars in spatial coincidence with neutrinos detected by IceCube. Specifically, we intended to investigate peculiar radio properties of the candidate counterparts related to neutrino production, as radio flares coincident to the neutrino detection or features in jets morphology (limb brightening, transverse structures). Methods. We collected multi-frequency VLBI follow-up observations of candidate counterparts of four high-energy neutrino events detected by IceCube between January 2019 and November 2020. We analysed their radio characteristics soon after the neutrino arrival in comparison with archival VLBI observations and low-frequency radio observations. We discussed our results with respect to previous statistical works and studies on the case of TXS 0506+056. Results. We identified and analysed in detail five potential neutrino emitting blazars. Our results suggest an enhanced state of radio activity for one source, PKS1725+123. However, the lack of adequate monitoring prior to the neutrino events was a limitation in tracing radio activity and morphological changes in all the sources. Conclusions. We suggest that PKS1725+123 is a promising neutrino source candidate. For the other four sources, our results alone do not allow us to reveal a strong connection between the radio activity state at neutrino arrival., Comment: 22 pages, 6 figures, accepted for publication in A&A