TANAMI is a multiwavelength program monitoring active galactic nuclei (AGN) south of -30{deg} declination including high-resolution Very Long Baseline Interferometry (VLBI) imaging, radio, optical/UV, X-ray and {gamma}-ray studies. We have previously published first-epoch 8.4GHz VLBI images of the parsec-scale structure of the initial sample. In this paper, we present images of 39 additional sources. The full sample comprises most of the radio- and {gamma}-ray brightest AGN in the southern quarter of the sky, overlapping with the region from which high-energy (>100TeV) neutrino events have been found. We characterize the parsec-scale radio properties of the jets and compare with the quasi-simultaneous Fermi/LAT {gamma}-ray data. Furthermore, we study the jet properties of sources which are in positional coincidence with high-energy neutrino events as compared to the full sample. We test the positional agreement of high-energy neutrino events with various AGN samples. TANAMI VLBI observations at 8.4GHz are made with Southern-Hemisphere radio telescopes located in Australia, Antarctica, Chile, New Zealand, and South Africa. Our observations yield the first images of many jets below -30{deg} declination at milliarcsecond resolution. We find that {gamma}-ray loud TANAMI sources tend to be more compact on parsec-scales and have higher core brightness temperatures than {gamma}-ray faint jets, indicating higher Doppler factors. No significant structural difference is found between sources in positional coincidence with high-energy neutrino events and other TANAMI jets. The 22 {gamma}-ray brightest AGN in the TANAMI sky show only a weak positional agreement with high-energy neutrinos demonstrating that the >100TeV IceCube signal is not simply dominated by a small number of the {gamma}-ray brightest blazars. Instead, a larger number of sources have to contribute to the signal with each individual source having only a small Poisson probability for producing an event in multi-year integrations of current neutrino detectors.