Structural steel plates oriented to resist shear loading can be used as hysteretic dampers in seismic force resisting systems. Some steel plate hysteretic dampers have engineered cut-outs leaving shear links that exhibit controlled yielding. A promising type of link described in the literature is the butterfly-shaped link that better aligns bending strength along the link length with the shape of the applied moment diagram. In previous studies, it has been observed that these links are capable of substantial ductility and energy dissipation. However, the effect of varying butterfly geometric parameters on the location of plastic hinges, accumulation of plastic strain, the potential for fracture, buckling, and energy dissipation are not well understood and thus deserve further investigation. A parametric computational study is conducted to investigate the shear yielding, flexural yielding, and lateral torsional buckling limit states for butterfly-shaped links. After validating the accuracy of the finite element (FE) modeling approach against previous experiments, 112 computational models with different geometrical properties were constructed and analyzed including consideration of initial imperfections. The resulting yielding moment, corresponding critical shear force, the accumulation of plastic strains through the length of links as well as the amount of energy dissipated are investigated. The results indicate that as the shape of the butterfly-shaped links become too straight or conversely too narrow in the middle, peak accumulated plastic strains increase. The significant effect of plate thickness on the buckling limit state is examined in this study. Results show that overstrength for these links (peak force divided by yield force) is between 1.2 - 4.5, with straight links producing larger overstrength. Additionally, proportioning the links to delay buckling, and designing the links to yield in the flexural mode are shown to improve energy dissipation.