We study the decay $\bar{B}^{0}\rightarrow\Lambda_{c}^{+}\bar{p}\pi^{+}\pi^{-}$, reconstructing the \Lambda_{c}^{+} baryon in the $p K^{-}\pi^{+}$ mode, using a data sample of $467\times 10^{6}$ $B\bar{B}$ pairs collected with the BaBar detector at the PEP-2 storage rings at SLAC. We measure branching fractions for decays with intermediate $\Sigma_{c}$ baryons to be ${\cal B}\bar{B}^{0}\rightarrow\Sigma_{c}(2455)^{++}\bar{p}\pi^{-}=(21.3 \pm 1.0 \pm 1.0 \pm 5.5) \times 10^{-5}$, ${\cal B}\bar{B}^{0}\rightarrow\Sigma_{c}(2520)^{++}\bar{p}\pi^{-}=(11.5\pm 1.0 \pm 0.5 \pm 3.0)\times 10^{-5}$, ${\cal B}\bar{B}^{0}\rightarrow\Sigma_{c}(2455)^{0}\bar{p}\pi^{+}=(9.1 \pm 0.7 \pm 0.4 \pm 2.4)\times10^{-5}$, and ${\cal B}\bar{B}^{0}\rightarrow\Sigma_{c}(2520)^{0}\bar{p}\pi^{+}= (2.2 \pm 0.7 \pm 0.1\pm 0.6) \times 10^{-5}$, where the uncertainties are statistical, systematic, and due to the uncertainty on the $\Lambda_{c}^{+}\rightarrow\proton\Km\pi^{+}$ branching fraction, respectively. For decays without $\Sigma_{c}(2455)$ or $\Sigma_{c}(2520)$ resonances, we measure ${\cal B}\bar{B}^{0}\rightarrow\Lambda_{c}^{+}\bar{p}\pi^{+}\pi^{-}_{\mathrm{non-\Sigma_{c}}}=(79 \pm 4 \pm 4 \pm 20)\times10^{-5}$. The total branching fraction is determined to be ${\cal B}\bar{B}^{0}\rightarrow\Lambda_{c}^{+}\bar{p}\pi^{+}\pi^{-}_{\mathrm{total}}=(123 \pm 5 \pm 7 \pm 32)\times10^{-5}$. We examine multibody mass combinations in the resonant three-particle $\Sigma_{c}\bar{p}\pi$ final states and in the four-particle $\Lambda_{c}^{+}\bar{p}\pi^{+}\pi^{-}$ final state, and observe different characteristics for the $\bar{p}\pi$ combination in neutral versus doubly-charged $\Sigma_{c}$ decays.