Extraction and recovery of palladium continues to be a technologically and economically challenging task, particularly in the recycling process of palladium from scrap automotive catalytic converters. In this work, extraction and transport of palladium(II) from HCl media with and without KSCN in the feed and/or in the stripping solutions by tris(2-ethylhexyl) phosphate (TEHP) and tris(2-ethylhexyl) phosphate (TEHP)-triisobutylphosphine sulfide (CYANEX 471X) mixture were studied in two- (liquid and membrane extraction) or three-phase (polymer inclusion membrane, PIM) configurations. Regarding PIMs, the efficiency factors (permeability, selectivity, and stability) were determined. Numerical data analysis revealed that the transport of palladium is achieved by a complex mechanism in which the solvating effects of TEHP and TEHP-CYANEX 471X are coupled to chloride ion co-transport and thiocyanic acid counter-transport. Palladium recovery from a spent automotive catalytic converter was accomplished (approximately 90%) with excellent separation from Pt(IV), Fe(III), Al(III), and other components via the studied membrane systems.