We consider parsimonious modeling of high-dimensional multivariate additive models (MAM) using regression splines, with or without sparsity assumptions. The approach is based on treating the coefficients as a third-order tensor and a Tucker decomposition is used to reduce the number of parameters in the tensor. The method can avoid the statistical inefficiency caused by estimating a large number of nonparametric functions. We establish the convergence rate of the proposed estimator. Numerical examples are presented to demonstrate the advantages of the proposed novel approach.