A search is presented for massive narrow-width resonances in the mass range of 1$-$4.5 TeV, decaying into pairs of Higgs bosons (HH). The search uses proton-proton collision data at a center-of-mass energy of 13 TeV collected with the CMS detector at the CERN LHC during 2016$-$2018, corresponding to an integrated luminosity of 138 fb$^{-1}$. The analysis targets final states where one Higgs boson decays into a pair of bottom quarks and the other into a pair of tau leptons, X $\to$ HH $\to$$\mathrm{b\bar{b}}τ^+τ^-$. It uses a single large jet to reconstruct the H $\to$$\mathrm{b\bar{b}}$ decay, while the H $\to$$τ^+τ^-$ decay products can either be contained within a single large jet or appear as two isolated tau leptons. The observed data are consistent with standard model background expectations. Upper limits at 95% confidence level are set on the production cross section for resonant HH production for masses between 1 and 4.5 TeV. This analysis sets the most sensitive limits to date on X $\to$ HH $\to$ $\mathrm{b\bar{b}}τ^+τ^-$ decays in the mass range of 1.4 to 4.5 TeV.