Parallel bicontinuous flows, which include stratified and core-annular flow, have applications in liquid-liquid extraction in microchannels. The flow regime has a significant impact on interphase mass transfer. Either stratified flow or core-annular flow can result in better extraction, depending on the physical properties of the fluids and solute and the operating conditions. In this work, we systematically compare the extraction performance of core-annular and stratified flow. Mathematical models are developed for each flow regime and solved semianalytically. Both models are validated with experimental data from the literature. Using the models we analyze both flow regimes across the parameter space. Two basis for comparison are used: (i) specified flow rates of the two fluid streams and (ii) specified pressure gradient and holdup (volume fraction of the carrier stream). For core-annular flow, two distinct cases are analyzed based on the position of the solute bearing carrier stream: (i) the carrier stream is the core fluid and the solvent stream is the annular fluid and (ii) the carrier stream is the annular fluid. The results are explained in terms of two key factors: the interfacial area and the ratio of diffusion time to residence time in each fluid. This new understanding is coalesced into a set of guidelines for selecting the parallel flow regime which enhances extraction performance.