We examine the production of neutral current Drell-Yan (DY) lepton pairs within the framework of the Standard Model Effective Field Theory (SMEFT). Utilizing the open-source fit platform $\texttt{xFitter}$, we explore the impact of high-statistics measurements of the neutral current DY (NCDY) forward-backward asymmetry $A_{\rm{FB}}$ near the weak boson mass scale in both the present and upcoming stages of the Large Hadron Collider (LHC). Our analysis not only confirms previous findings regarding the sensitivity of $A_{\rm{FB}}$ to parton distribution functions (PDFs) but also delves into the precise determination of $Z$-boson couplings to left-handed and right-handed $u$-quarks and $d$-quarks. Additionally, we investigate contributions beyond the Standard Model using the SMEFT framework. We provide insights into the significance of the $A_{\rm{FB}}$ asymmetry for the electroweak SMEFT fit and the precision study of $Z$-boson physics at both the LHC and the high-luminosity HL-LHC.