In this work, we explore a novel search strategy focusing on the decay channel $h \to Z a \to \ell\ell \tau\tau$, in which the Higgs boson decays to a $Z$ boson (with leptonic final states) and a light Beyond Standard Model (BSM) pseudoscalar $a$ subsequently decaying into a $\tau$-lepton pair. After validating our analysis framework against existing ATLAS and CMS searches in related exotic Higgs channels, we project the sensitivity of this search at the High-Luminosity Large Hadron Collider (HL-LHC), specifically targeting the two-body kinematic configuration of the Higgs decay. Our results demonstrate sensitivity to pseudoscalar masses in the range $m_a \in [5, 33]$ GeV, by exploiting both leptonic and hadronic $\tau$ decays. We present model-independent 95$\%$ C.L. limits on the branching ratio ${\rm BR}(h \to Z a) \times {\rm BR}(a \to \tau\tau)$, which establish $a \to \tau\tau$ as a powerful probe of light pseudoscalars. In certain well-motivated scenarios, the sensitivity obtained is stronger than that from existing projections in the $a \to \mu\mu$ and $a \to \gamma\gamma$ final states. Finally, we assess the implications of this search for an Axion-Like Particle (ALP) explanation of the muon $(g-2)_{\mu}$ anomaly, showing that the proposed $h \to Z a$, $a \to \tau\tau$ channel can set complementary constraints to those derived from $h \to Z a$, $a \to \gamma\gamma$ searches.