We report tunneling phenomena in double In$_{0.53}$Ga$_{0.47}$As quantum-well structures that are at odds with the conventional parallel-momentum-conserving picture of tunneling between two-dimensional systems. We found that the tunneling current was mostly determined by the correlation between the emitter and the state in one well, and not by that between those in both wells. Clear magnetic-field-dependent features were first observed before the main resonance, corresponding to tunneling channels into the Landau levels of the well near the emitter. These facts provide evidence of the violation of in-plane momentum conservation in two-dimensional systems.