In the finite element method (FEM), the solutions of Partial Differential Equations (PDEs) are approximated using linear combinations of prescribed basis functions. The coefficients of the linear combinations are obtained by solving a system of linear equations. The FEM allows for the solution of a PDE for fixed values of the PDE parameters. It is not possible to obtain “at once” the family of solutions of parametric PDEs using FEM. We proposed to introduce B-spline basis functions into neural network approximations, where the coefficients of the basis functions used to approximate the solution are predicted by a neural network. Direct approximation of B-spline coefficients by NN has several advantages compared to standard FEM. First, it allows us to obtain a family of solutions of the parametric PDE “at once”. The PDE parameters are input to the neural network, and the output involves the coefficients of the basis functions. Second, it allows obtaining the solution of a parametric PDE without the construction and solution of a system of linear equations. Third, since neural networks are universal approximators, direct approximation of B-spline coefficients by NN may find a dependence between the PDE parameters and the coefficients of the basis functions used to approximate the solution. The training of our method requires learning the dependence between the PDE parameters and the basis functions’ coefficients. The approximations of B-spline coefficients by NN inherit all the features of standard FEM approximations.