In this paper, we explore a new version of dark energy called Barrow holographic dark energy within the framework modified gravity called $f (Q, T)$ gravity by adopting the simple homogeneous, isotropic, and spatially flat Friedmann–Robertson–Walker (FRW) model of the universe. Our goal is to understand how the universe evolved over time. To do this, we use parameterizetion of Hubble’s parameter method. We then use a powerful tool called Monte-Carlo Markov Chain to find the best values for the constants in our formula. We do this by comparing our formula to actual data from observations of the universe. Once we have the best values for the constants, we calculate other important parameters that describe the universe’s evolution. These include: Deceleration parameter which measures how quickly the expansion is slowing down. We found $q_{0} = - 0.60 1_{- 0.0131}^{+ 0.0131}$ . Equation of state parameter measures the properties of dark energy. We find $ω_{0} = - 0.701 8_{- 0.0101}^{+ 0.0101}$ . We also study the stability and energy conditions along with the state-finder and $O_{m} (z)$ -parameter of our model to ensure it is consistent with our understanding of the universe.