Abstract: The parametric finite difference method of lines is an effective numerical method for solving the partial differential equations in the domains with curved boundaries. However, for the degenerate lines on the boundary, how rationally to deal with required boundary conditions remains an open problem. This paper resolves the problem and “finds out” the needed boundary conditions on degenerate lines. Taking a two-dimensional Poisson equation as an example, the detailed derivation of related formulas is presented, and the validity, rationality, and accuracy of derived boundary conditions are demonstrated by the numerical results of computational examples.