<p>In this paper, two classes of efficient Laplace-modified Galerkin quadratic spline schemes for parabolic problems with time-dependent variable coefficients are presented, where the Laplace-modified technique, together with first/second-order temporal discretization and Galerkin quadratic spline spatial discretization are employed. The resulting linear algebraic system to be solved per time level is shown owning the same constant-coefficient matrix structure, which is time-independent and thus can be generated only once and in advance. That is to say, the proposed schemes are easy to implement and are much more computationally efficient than other well-known schemes for the modeling of time-dependent problems with variable coefficients. Furthermore, the presented schemes are shown to be unconditionally stable under suitable choice on the stabilization parameter. Optimal-order error estimates in energy norm are also rigorously proven for the schemes. Finally, ample numerical experiments are carried out to verify the accuracy, efficiency and effectiveness in the modeling of linear and nonlinear parabolic problems with variable coefficients, or even nonlinear coefficients and mixed spatial derivative terms.</p>