We observed with the Very Long Baseline Array (VLBA) at 2.3 and 8.6 GHz a complete flux-density-limited sample of 482 radio sources with decl. > +75° brighter than 200 mJy at 1.4 GHz drawn from the NVSS catalog. A total of 34% of the sources show parsec-scale emission above the flux density detection limit of 30 mJy; their accurate positions and parsec-scale structure parameters are determined. Among all the sources detected at least at the shortest VLBA baselines, the majority, or 72%, have a steep single-dish spectrum. The fraction of the sources with a detectable parsec-scale structure is above 95% among the flat-spectrum objects and close to 25% among the steep-spectrum objects. We identified 82 compact steep-spectrum source candidates, which make up 17% of the sample; most of them are reported for the first time. The compactness and the brightness temperature of the sources in our sample show a positive correlation with single-dish and VLBA spectral indices. All the sources with a significant 8 GHz variability were detected by the VLBA snapshot observations, which independently confirmed their compactness. We demonstrated that 54% of the sources detected by the VLBA at 2.3 GHz in our sample have a steep VLBA spectrum. The compact radio emission of these sources is likely dominated by optically thin jets or mini-lobes, not by an opaque jet core. These results show that future VLBI surveys aimed at searching for new sources with parsec-scale structure should include not only flat-spectrum sources but also steep-spectrum ones in order to reach an acceptable level of completeness.