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The influence of parametric forcing on a viscoelastic fluid layer, in both gravitationally stable and unstable configurations, is investigated via linear stability analysis. When such a layer is vertically oscillated beyond a threshold amplitude, large interface deflections are caused by Faraday instability. Viscosity and elasticity affect the damping rate of momentary disturbances with arbitrary wavelength, thereby altering the threshold and temporal response of this instability. In gravitationally stable configurations, calculations show that increased elasticity can either stabilize or destabilize the viscoelastic system. In weakly elastic liquids, higher elasticity increases damping, raising the threshold for Faraday instability, whereas the opposite is observed in strongly elastic liquids. While oscillatory instability occurs in Newtonian fluids for all gravity levels, we find that parametric forcing below a critical frequency will cause a monotonic instability for viscoelastic systems at microgravity. Importantly, in gravitationally unstable configurations, parametric forcing above this frequency stabilizes viscoelastic fluids, until the occurrence of a second critical frequency. This result contrasts with the case of Newtonian liquids, where under the same conditions, forcing stabilizes a system for all frequencies below a single critical frequency. Analytical expressions are obtained under the assumption of long wavelength disturbances predicting the damping rate of momentary disturbances as well as the range of parameters that lead to a monotonic response under parametric forcing.