A search for a pair of low-mass pseudoscalars a that promptly decay into τ-leptons is presented using 140 fb^-1 of proton-proton collision data at 13 TeV centre-of-mass energy recorded with the ATLAS detector at the Large Hadron Collider. The result is used to place constraints on exotic decays of the Higgs boson into four τ-leptons, H → aa → τ⁺τ^-τ„⁺τ„^-. This search focuses on events with either one or two tau-leptons decaying into hadrons and neutrinos, and the remaining three or two τ-leptons decaying into either electron or muon and neutrinos. No significant excess is observed above the expected Standard Model background and upper limits at the 95% confidence level on B(H → aa → τ⁺τ^-τ^„+τ„^-) are set ranging from 0.06 to 0.23, depending on the mass m_a ranging from 15 to 60 GeV.