The approximation of point clouds in terms of parametric representations is a fundamental task for geometric modeling and processing applications to properly analyze the (re-)constructed model in its full complexity. A necessary and key step in this process requires the identification of a suitable data parameterization. If the data connectivity is determined by a rectilinear grid, standard closed-form methods are available for general multivariate configurations. Existing data-driven parameterization methods instead are limited to the univariate case and require pre/post-processing steps of the input data to handle point sequences without fixed length. In this paper we propose a dimension independent method based on convolutional neural networks to assign parameter values to gridded point clouds of arbitrary size, without the need for additional data processing steps. We train the proposed networks by considering polynomial least squares approximations and demonstrate, both in the univariate and bivariate settings, that the accuracy of the final model properly scales when uniform and adaptive spline refinement is considered. A selection of numerical experiments on point clouds of different sizes highlights the performance of our parameterization scheme. Noisy data sets which simulate measurement errors are also considered.