This study investigates the $\beta $ -decay properties of $sd$ -shell nuclei using the proton-neutron quasiparticle random phase approximation (pn-QRPA) model. The computed Gamow-Teller (GT) strength distributions show decent agreement with the measured data. The calculated $\beta $ -decay half-lives show good agreement with the previous shell model calculations. The computed log $\textit{ft}$ values align well with the available experimental data, validating the consistency of the theoretical approach. A key advancement of this work is the calculation of stellar weak interaction rates performed without assuming the Brink-Axel hypothesis for the estimation of GT distributions from parent excited states. The sum of $\beta ^{-}$ and positron capture ( $\beta ^{-}$ + PC) rates were compared with earlier predictions from the shell model. The percentage contribution of $\beta ^{-}$ and PC is also investigated under stellar conditions. At low density and high temperature ( $\rho =10^{7}\text{ g}/\text{cm}^{3}$ , T = 30 GK) the pn-QRPA calculation compare well with the shell model and differs at most by a factor 10. Our findings may provide crucial and refined nuclear inputs for astrophysical simulations of $r$ - and $s$ -process nucleosynthesis.
