A study of the charge conjugation and parity ($\textit{CP}$) properties of the interaction between the Higgs boson and $\tau $-leptons is presented. The study is based on a measurement of $\textit{CP}$-sensitive angular observables defined by the visible decay products of $\tau $-leptons produced in Higgs boson decays. The analysis uses 139 fb$^{-1}$ of proton–proton collision data recorded at a centre-of-mass energy of $\sqrt{s}= 13$ TeV with the ATLAS detector at the Large Hadron Collider. Contributions from $\textit{CP}$-violating interactions between the Higgs boson and $\tau $-leptons are described by a single mixing angle parameter $\phi _{\tau }$ in the generalised Yukawa interaction. Without constraining the $H\rightarrow \tau \tau $ signal strength to its expected value under the Standard Model hypothesis, the mixing angle $\phi _{\tau }$ is measured to be $9^{\circ } \pm 16^{\circ }$, with an expected value of $0^{\circ } \pm 28^{\circ }$ at the 68% confidence level. The pure $\textit{CP}$-odd hypothesis is disfavoured at a level of 3.4 standard deviations. The results are compatible with the predictions for the Higgs boson in the Standard Model.