In this paper, we investigate the evolution of the FLRW universe by analyzing the constant jerk parameter $j$ within the framework of $f(T)$ gravity. Using a model-independent parametrization approach, we assume $j$ as constant and derive an expression for the Hubble parameter as \(\begin{aligned} H(z)=H_{0}\left [\xi _{1}\left (z+1\right )^{\frac{3+\sqrt{8j+1}}{2}}+ \left (1{-\xi }_{1}\right )\left (z+1\right )^{\frac{3-\sqrt{8j+1}}{2}} \right ]^{\frac{1}{2}} \end{aligned}\) We estimate model parameters via a Chi-square test coupled with Markov Chain Monte Carlo (MCMC) simulations, based on 52 Observational Hubble Data (OHD) points and 1701 Pantheon+SHOES data points. This method yields the best fit values: $\xi _{1}={0.39}_{-0.08}^{+0.12}$ , $j={0.67}_{-0.19}^{+0.19}$ , $H_{0}={66.08}_{-2.78}^{+2.83} \; km/s/Mpc$ for OHD dataset and $\xi _{1}={0.37}_{-0.03}^{+0.03}$ , $j={0.67}_{-0.38}^{+0.40}$ , $H_{0}={72.80}_{-0.29}^{+0.31}$ $km/s/Mpc$ for Pantheon+SHOES dataset. The value of $H_{0}$ obtained in our analysis closely resembles the value estimated by the Planck Collaboration in 2018, $H_{0}=67.4\pm 0.5\; km/s/Mpc$ . Our analysis of the deceleration parameter $q$ reveals a transition from an early decelerating phase to a present accelerating expansion, with $q_{0}\approx -0.3887$ $(OHD)$ and $q_{0}\approx -0.4139$ $(Pantheon+SHOES)$ . Additionally, we examine dynamic parameter such as energy density, pressure, and equation of state parameter within the model $f\left (T\right )=T+\eta T^{\beta }$ , where $\eta $ and $\beta $ are an arbitrary real constant. The equation of state parameter $\omega $ in our model demonstrates a smooth transition from the radiation-dominated era to the matter-dominated era and finally to the dark energy era. The present-day values of $\omega _{0} \approx -0.6196$ $(OHD)$ and $\omega _{0} \approx -0.6548$ $(Pantheon+SHOES)$ suggests that the universe is currently in a quintessence phase, where dark energy behaves as a dynamical component rather than a cosmological constant $\omega = -1$ . This behavior indicates that our model is physically acceptable and aligns with the general features of cosmic evolution. Notably, the strong energy condition (SEC) is violated at present $\left (z =0\right )$ and is projected to remain violated in the future $\left (z < 0\right )$ , driving the observed accelerated expansion of the universe. Also, this study offers insights into the impact of a constant jerk parameter on cosmic evolution and expansion dynamics within the framework of modified gravity, with a specific focus on the $f\left (T\right )$ gravity model.
