Understanding the role of multi-dimensional conjugate heat transfer on the thermal efficiency of mini/micro counterflow heat exchangers is critical for the design of high efficient equipment. This topic is addressed here by considering a simple model with exact solution: the laminar counterflow parallel-plate heat exchanger. Using as starting point the eigenfunction series solution recently obtained, by the authors, a thorough parametric study is carried out to investigate the role of the two dimensionless parameters involved in multi-dimensional wall conduction: the dimensionless wall thickness, Delta(w), and the dimensionless wall thermal resistance, K-w(-1). The analytical eigencondition is first presented and discussed, and the associated eigenvalue spectrum is analyzed using contour plots of the lowest-order eigenvalues in the (Delta(w), K-w(-1)) plane. The complex task of determining the eigenvalues numerically is largely facilitated by approximate expressions obtained from the asymptotic analysis of the singularities that appear in the eigencondition. The fast evaluation of the eigenvalues makes it possible to obtain contour plots of the heat exchanger effectiveness in the (Delta(w), K-w(-1)) plane, which exhibit distinguished regimes corresponding to limiting cases with and without axial and transverse wall conduction effects, with smooth transitions occurring for moderately small values of Delta(w) and K-w(-1). The analysis provides conditions for neglecting axial and transverse wall conduction, and shows that an optimum wall conductivity always exists in heat exchangers with sufficiently thin walls. (C) 2017 Elsevier Ltd. All rights reserved.