We study the rare decays $B_u^{+}\to {\pi }^{+}{\ell }^{+}{\ell }^{-}$, ${\rho }^{+}{\ell }^{+}{\ell }^{-}$ and $B_d^0\to {\ell }^{+}{\ell }^{-}$ ($\ell =e$, $\mu$) in the $R$-parity violating supersymmetric standard model. From the latest upper limits of $\mathcal{B}(B_u^{+}\to {\pi }^{+}{\ell }^{+}{\ell }^{-})$ and $\mathcal{B}(B_d^0\to {\ell }^{+}{\ell }^{-})$, we have derived new upper bounds on the relevant $R$-parity violating couplings products, which are stronger than the existing ones. Using the constrained parameter space, we present the $R$-parity violating effects on the branching ratios and the forward-backward asymmetries of these decays. We find that $\mathcal{B}(B_d^0\to {\ell }^{+}{\ell }^{-})$ and $\mathcal{B}(B_u^{+}\to {\rho }^{+}{\ell }^{+}{\ell }^{-})$ could be enhanced several orders by the $R$-parity violating sneutrino and squark exchanges, respectively. The $R$-parity violating effects on the dilepton invariant mass spectra of $B_u^{+}\to {\pi }^{+}{\ell }^{+}{\ell }^{-}$, ${\rho }^{+}{\ell }^{+}{\ell }^{-}$ and the normalized forward-backward asymmetry ${\mathcal{A}}_{\mathrm{FB}}(B_u^{+}\to {\pi }^{+}{\ell }^{+}{\ell }^{-})$ and ${\mathcal{A}}_{\mathrm{FB}}(B_u^{+}\to {\rho }^{+}{\ell }^{+}{\ell }^{-})$ are studied in detail. Our results could be used to probe the $R$-parity violating effects and will correlate with searches for the direct $R$-parity violating signals at the future experiments.

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