The problem of elastic antiplane with a parabolic crack was investigated. The exterior region of the parabolic crack was mapped out of a unit circle through conformal transformation. A boundary integral method was proposed to avoid the difficulty caused by the singularities of the transform function. The complex potential solution to the elastic antiplane boundary value problem with a parabolic crack was obtained. The stress intensity factor (SIF) at the tip of parabolic crack was thus given by employing a formula proposed in the present article to calculate SIF directly on the basis of complex potentials. The solution can be reduced to the solution for line crack in a special case. The shape of the parabolic crack and the ratio of the shear stress loads of two directions at infinity affect the magnitude of stress intensity factors.