We study efficient surrogate approximations of parametric forward operators arising in simulations of brain tumor growth governed by systems of semi-linear parabolic partial differential equations. The mathematical model consists of coupled reaction–diffusion equations on heterogeneous two- and three-dimensional domains, with state variables representing tumor cell populations, oxygen concentration, and tissue compartments. Model dynamics are driven by nonlinear reaction terms, spatially varying diffusion coefficients, and a moderately high-dimensional parameter vector encoding biological and physiological uncertainties. After spatial discretization, the governing equations yield large-scale systems of nonlinear ordinary differential equations. Time integration is performed using operator-splitting schemes that decouple diffusion and reaction processes. The resulting full-order model (FOM) defines a nonlinear, time-dependent parameter-to-state map whose repeated evaluation is computationally prohibitive in many-query settings such as inverse problems, statistical inference, or digital twin applications. To address this challenge, we construct surrogate models based on reduced-order modeling (ROM) of the discrete forward operator. We investigate several tensorial ROM variants, using a standard proper orthogonal decomposition ROM as a baseline. Particular attention is given to the influence of parameter dimensionality, nonlinear effects, and spatio-temporal coupling on approximation accuracy and computational efficiency. The surrogate models are trained on ensembles of high-fidelity simulations over structured parameter samples and are evaluated in both two- and three-dimensional settings. Model performance is assessed by comparing reconstruction errors and runtimes against the FOM across a range of parameter values. The results demonstrate that substantial computational savings can be achieved while maintaining high fidelity to reference solutions, indicating that the proposed framework is well-suited for many-query settings.