Context. The advent of Fermi is changing our understanding on the radio and γ-ray emission in active galactic nuclei. Unlike preFermi ideas, BL Lac objects are found to be the most abundant emitters in the γ-ray band. However, since they are relatively weak radio sources, most of their parsec-scale structure and their multifrequency properties are poorly understood and/or have not been investigated in a systematic fashion. Aims. Our main goal is to analyze the radio and γ-ray emission properties of a sample of 42 BL Lacs selected, for the first time in the literature, with no constraint on their radio and γ-ray emission. Methods. Thanks to new Very Long Baseline Array observations at 8 and 15 GHz for the whole sample, we present fundamental parameters here such as radio flux densities, spectral index information, and parsec-scale structure. Moreover, we search for γ-ray counterparts using data reported in the Second Catalog of Fermi Gamma-ray sources. Results. Parsec-scale radio emission is observed in the majority of the sources at both frequencies. Gamma-ray counterparts are found for 14/42 sources. Conclusions. The comparison between our results in radio and gamma-ray bands points out the presence of a large number of faint BL Lacs showing "non-classical" properties such as low source compactness, core dominance, no gamma-ray emission, and steep radio spectral indexes. A deeper multiwavelength analysis will be needed.