Observational information on high-energy astrophysical neutrinos is being continuously collected by the IceCube observatory. However, the sources of the neutrinos are still unknown. In this study we use radio very-long-baseline interferometry (VLBI) data for a complete VLBI-flux-density limited sample in a statistical manner in order to address the problem of the origin of astrophysical neutrinos with energies above 200 TeV. We find that active galactic nuclei (AGN) positionally associated with IceCube events have typically stronger parsec-scale cores than the rest of the sample, with the post-trial probability of chance coincidence of 0.2%. We select the four strongest AGN as highly probable associations: 3C 279, NRAO 530, PKS 1741-038, and OR 103. Moreover, we find the increase of radio emission at frequencies above 10 GHz around neutrino arrival times for several other VLBI-selected AGN on the basis of RATAN-600 monitoring. The most pronounced example of such behaviour is PKS 1502+106. We conclude that AGN with bright Doppler-boosted jets constitute an important population of neutrino sources. High-energy neutrinos are produced in their central parsec-scale regions, probably in proton-photon interactions at or around the accretion disk. Radio-bright AGN likely associated with neutrinos have very diverse gamma-ray properties suggesting that gamma rays and neutrinos may be produced in different regions of the AGN and not directly related. A small viewing angle of the jet-disk axis is however required to detect either of them.