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The competition between intertube hopping processes and density-density interactions is investigated in one-dimensional quantum dipolar bosons systems of A' coupled tubes at zero temperature. Using a phenomenological bosonization approach, we show that the resulting competition leads to an exotic quantum phase transition described by a U(l) x Z/v conformai field theory with a fractional central charge. The emerging Zi parafermionic critical degrees of freedom are highly nontrivial in terms of the original atoms or polar molecules of the model. We further determine the main physical properties of the quantum-critical point in a double-tube system which has central charge c = 3/2. In triple-tube systems, we show that the competition between the two antagonistic processes is related to the physics of the two-dimensional Z3 chiral Potts model. This work opens the possibility to study the exotic properties of the Z parafermions in the context of ultracold quantum Bose gases. [ABSTRACT FROM AUTHOR]