We present a finite elements-neural network approach for the numerical approximation of parametric partial differential equations. The algorithm generates training data from finite element simulations, and uses a data -driven (supervised) feedforward neural network for the online approximation of the solution. The objective is to ensure that the overall error of the method is below some preset tolerance, and we thus control and balance the error coming from the finite element method, and the one introduced by the neural network approximation. Two finite element methods are considered and compared; a fixed grid approach uses the same mesh for all values of the parameters, while an adaptive finite elements approach enforces the same discretization error uniformly in the parameters space. Numerical results are presented for an elliptic model problem. The fixed grid approach shows limitations in terms of error balancing and control, while the adaptive approach allows a better accuracy and more flexibility of the method. We conclude by proposing an adaptive algorithm to control the size of the training set given a network architecture and ensure that the overall error of the method is below a given tolerance.