In this study, a parametric intuitionistic fuzzy multi-objective fractional transportation problem (PIF-MOFTP) is proposed. The current PIF-MOFTP has a single-scalar parameter in the objective functions and an intuitionistic fuzzy supply and demand. Based on the $\left(\alpha ,\beta \right)$-cut concept a parametric $\left(\alpha ,\beta \right)$-MOFTP is established. Then, a fuzzy goal programming (FGP) approach is utilized to obtain $\left(\alpha ,\beta \right)$-Pareto optimal solution. We investigated the stability set of the first kind (SSFK) corresponding to the solution by extending the Kuhn-Tucker optimality conditions of multi-objective programming problems. An algorithm to crystalize the progressing SSFK for PIF-MOFTP as well as an illustrative numerical example is presented.