Engineering problems are often defined depending on parameters. At the design stage for instance, the optimization process involves tuning a large number of parameters. Most development in the field of reduced order modelling involves the identification of latent parameters, i.e. the coordinates in a low-rank manifold, tangent to the high dimensional problem or defined nonlinearly. This strategy provides massive gains but still faces the problem of explicit parametric dependency, when the natural language of user-defined parameters has to be translated into the language of ideal latent parameters. Due to the number of explicit parameters, such models still face a wall of complexity known as the curse of dimensionality. We propose a tensor regression framework tailored to scale with such parametric, nonlinear problems in scarce data contexts. Separation of variables provides us with an array of rank-1 tensors to build the regression operator. We leverage tensor algebra and identify kernel properties that allow us to compute parametric approximations, able to scale to hundreds of parameters while preserving well-known matrix-based properties. The fixed-point strategy usually encountered for such problems is avoided to produce a least squares solution without iterating. The approach is applied to a POD-based reduced model to demonstrate scaling and robustness of the approach. The approach is compared to the well-known sparse PGD to demonstrate the low complexity, efficiency and ease-of-use of our framework.