Air bubbles entrained in complex fluids pose persistent challenges across a wide range of industrial processes, including polymer processing, coating flows, and chemical reactors. Coupled with a complex rheological property, such as viscoelasticity, the rising dynamics of air bubble deviate markedly from their Newtonian counterparts, exhibiting phenomena such as velocity jumps, negative wakes, and pronounced shape deformations. These behaviors arise from the strong nonlinear coupling between flow kinematics marking predictive understanding particularly difficult. As bubble-induced defects can critically degrade performance of process, a systematic and mechanistic analysis of bubble motion in viscoelastic multiphase systems is essential. To address this challenge, we perform a computational investigation of a rising air bubble in a viscoelastic liquid using a Phase Field framework implemented within a Finite-Element formulation. The governing equations for the coupled multiphase and viscoelastic flow are solved using FEniCS, an open-source finite element platform. Using this framework, we systematically vary key material parameters—including surface tension, relaxation time, and the retardation parameter (solvent to total viscosity ratio)—to elucidate their individual and combined effects on bubble dynamics. The computed bubble morphologies are classified into four representative regimes: elongation, continuous splitting, early-stage splitting, and quasi-spherical shapes. By quantitatively linking these transitions to the magnitude of viscoelastic stress and the spatial location of stress relaxation relative to the bubble interface, we reveal how viscoelasticity governs both rising velocity and morphological variation. These results provide clear mechanistic insights into bubble behavior in viscoelastic liquids and offer guidance in industrial applications