A comprehensive census of Galactic open cluster properties places unique constraints on the Galactic disc structure and evolution. In this framework, we investigate the evolutionary status of three poorly studied open clusters, Berkeley 31, Berkeley 23 and King 8, all located in the Galactic anticentre direction. To this aim, we make use of deep Large Binocular Telescope observations, reaching more than 6mag below the main-sequence turn-off. To determine the cluster parameters, namely age, metallicity, distance, reddening and binary fraction, we compare the observational colour--magnitude diagrams (CMDs) with a library of synthetic CMDs generated with different evolutionary sets (Padova, FRANEC and FST) and metallicities. We find that Berkeley 31 is relatively old, with an age between 2.3 and 2.9Gyr, and rather high above the Galactic plane, at about 700pc. Berkeley 23 and King 8 are younger, with best-fitting ages in the range 1.1-1.3 and 0.8-1.3 Gyr, respectively. The position above the Galactic plane is about 500-600 pc for the former and 200 pc for the latter. Although a spectroscopic confirmation is needed, our analysis suggests a subsolar metallicity for all three clusters. Based on observations collected at the Large Binocular Telescope (LBT) and in part at the Italian Telescopio Nazionale Galileo (TNG). The LBT is an international collaboration among institutions in the United States, Italy and Germany. LBT Corporation partners are: The University of Arizona on behalf of the Arizona University system; Istituto Nazionale di Astrofisica, Italy; LBT Beteiligungsgesellschaft, Germany, representing the Max-Planck Society, the Astrophysical Institute Potsdam, and Heidelberg University; The Ohio State University, and The Research Corporation, on behalf of The University of Notre Dame, University of Minnesota and University of Virginia. The TNG is operated on the island of La Palma by the Fundacion Galileo Galilei of the INAF (Istituto Nazionale di Astrofisica) at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias.