The Bologna complete sample of nearby radio sources II. Phase referenced observations of faint nuclear sources

Aims. To study statistical properties of different classes of sources, it is necessary to observe a sample that is free of selection effects. To do this, we initiated a project to observe a complete sample of radio galaxies selected from the B2 Catalogue of Radio Sources and the Third Cambridge Revised Catalogue (3CR), with no selection constraint on the nuclear properties. We named this sample the Bologna Complete Sample (BCS). Methods. We present new VLBI observations at 5 and 1.6 GHz for 33 sources drawn from a sample not biased toward orientation. By combining these data with those in the literature, information on the parsec-scale morphology is available for a total of 76 of 94 radio sources with a range in radio power and kiloparsec-scale morphologies. Results. The fraction of two-sided sources at milliarcsecond resolution is high (30%), compared to the fraction found in VLBI surveys selected at centimeter wavelengths, as expected from the predictions of unified models. The parsec-scale jets are generally found to be straight and to line up with the kiloparsec-scale jets. A few peculiar sources are discussed in detail.