A search for heavy neutral Higgs bosons is performed using the LHC Run 2 data, corresponding to an integrated luminosity of 139 fb^{-1} of proton-proton collisions at sqrt[s]=13 TeV recorded with the ATLAS detector. The search for heavy resonances is performed over the mass range 0.2-2.5 TeV for the τ^{+}τ^{-} decay with at least one τ-lepton decaying into final states with hadrons. The data are in good agreement with the background prediction of the standard model. In the M_{h}^{125} scenario of the minimal supersymmetric standard model, values of tanβ>8 and tanβ>21 are excluded at the 95% confidence level for neutral Higgs boson masses of 1.0 and 1.5 TeV, respectively, where tanβ is the ratio of the vacuum expectation values of the two Higgs doublets.