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We search for the isotropic stochastic gravitational-wave background,including the nontensorial polarizations that are allowed in general metrictheories of gravity, in the Parkes Pulsar Timing Array (PPTA) second datarelease (DR2). We find no statistically significant evidence that thecommon-spectrum process reported by the PPTA collaboration has the tensortransverse, scalar transverse, vector longitudinal, or scalar longitudinalcorrelations in PPTA DR2. Therefore, we place a $95\%$ upper limit on theamplitude of each polarization mode, as $\mathcal{A}_{\mathrm{TT}} \lesssim3.2\times 10^{-15}$, $\mathcal{A}_{\mathrm{ST}} \lesssim 1.8\times 10^{-15}$,$\mathcal{A}_{\mathrm{VL}}\lesssim 3.5\times 10^{-16}$ and$\mathcal{A}_{\mathrm{SL}}\lesssim 4.2\times 10^{-17}$; or, equivalently, a$95\%$ upper limit on the energy density parameter per logarithm frequency, as$\Omega_{\mathrm{GW}}^{\mathrm{TT}} \lesssim 1.4\times 10^{-8}$,$\Omega_{\mathrm{GW}}^{\mathrm{ST}} \lesssim 4.5\times 10^{-9}$,$\Omega_{\mathrm{GW}}^{\mathrm{VL}} \lesssim 1.7\times 10^{-10}$ and$\Omega_{\mathrm{GW}}^{\mathrm{SL}} \lesssim 2.4\times 10^{-12}$, at afrequency of 1/yr.

Constraining the Polarization of Gravitational Waves with the Parkes Pulsar Timing Array Second Data Release