We search for isotropic stochastic gravitational-wave background including non-tensorial polarizations allowed in general metric theories of gravity in the Parkes Pulsar Timing Array (PPTA) second data release (DR2). We find no statistically significant evidence that the common process reported by the PPTA collaboration has the tensor transverse (TT), scalar transverse (ST), vector longitudinal (VL), or scalar longitudinal (SL) correlations in PPTA DR2. Therefore, we place $95%$ upper limit on the amplitude of each polarization mode as $mathcal{A}_{mathrm{TT}} lesssim 3.2times 10^{-15}$, $mathcal{A}_{mathrm{ST}} lesssim 1.8times 10^{-15}$, $mathcal{A}_{mathrm{VL}}lesssim 3.5times 10^{-16}$ and $mathcal{A}_{mathrm{SL}}lesssim 4.2times 10^{-17}$; or equivalently, the $95%$ upper limit on the energy density parameter per logarithm frequency as $Omega_{mathrm{GW}}^{mathrm{TT}} lesssim 1.4times 10^{-8}$, $Omega_{mathrm{GW}}^{mathrm{ST}} lesssim 4.5times 10^{-9}$, $Omega_{mathrm{GW}}^{mathrm{VL}} lesssim 1.7times 10^{-10}$ and $Omega_{mathrm{GW}}^{mathrm{SL}} lesssim 2.4times 10^{-12}$ at frequency of 1/year.
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