Recently, microfluidics deformability cytometry has emerged as a powerful tool for high throughput mechanical phenotyping of large populations of cells. These methods characterize cells by their mechanical fingerprints through exerting hydrodynamic forces and monitoring the resulting deformation. These devices have shown great promise for label free cytometry, yet there is a critical need to improve their accuracy and reconcile any discrepancies with other methods, such as atomic force microscopy. In this study, we employ computational fluid dynamics simulations and uncover how the elasticity of frequently used carrier fluids such as methyl-cellulose dissolved in phosphate-buffered saline is significantly influential to the resulting cellular deformation. We conducted CFD simulations that are conventionally used within the deformability cytometry field, which neglect fluid elasticity. Subsequently, we incorporated a more comprehensive model that simulates the viscoelastic nature of the carrier fluid. A comparison of the predicted stresses between these two approaches underscores the significance of the emerging elastic stresses in addition to the well-recognized viscous stresses along the channel. Furthermore, we utilize a two-phase flow model to predict the deformation of a promyelocyte (i.e. HL-60 cell type) within a hydrodynamic constriction channel. The obtained results highlight a substantial impact of the elasticity of carrier fluid on cellular deformation and raise questions about the accuracy of mechanical property estimates derived by neglecting elastic stresses.