For a second order PDE $A u_{xx} + B u_{xt} + C u_{tt} + \dots = 0$, the discriminant is $B^2 - 4AC$.
Here $A = \alpha, B = 0, C = 0$, so $B^2 - 4AC = 0$, which classifies the heat equation as parabolic. (The wave equation is hyperbolic with $B^2 - 4AC > 0$, and Laplace equation is elliptic with $B^2 - 4AC < 0$).