Electron transport through a parallel double quantum dot is theoretically studied in the Kondo regime with the use of the non-equilibrium Green function formalism based on the equation of motion method. An influence of inter-dot tunnel coupling t and channel mixing effects on the orbital (spinless) Kondo phenomenon is analysed in the linear and nonlinear transport regimes. Both effects lead to a considerable suppression of the conductance in the Kondo regime. In a system with dots capacitively coupled (t = 0) the differential conductance shows a zero-bias peak whose intensity diminishes gradually with mixing effects included. When tunnel coupling between dots is taken into account () the orbital Kondo resonance splits and the intensities of both components are strongly influenced by channel mixing. The linear conductance calculated as a function of a dot level position E₀ is strongly suppressed in the Kondo regime but at higher values of E₀ a relatively well-pronounced side peak appears whose intensity increases with the coupling rate t. We consider the side peak maximum as originated from interference processes in the system.