Parameter identification is important in many engineering processes, and benefits from quick evaluations with reduced models. We present here a framework to solve inverse problems for parametric partial differential equations. The reduced model is built on a neural network method for the numerical approximation of a given parametric partial differential equation. Training data are generated thanks to adaptive finite element simulations. A supervised feedforward neural network is then used for the online approximation of the solution. The inverse problem aims at identifying parameters using available measurement data. The corresponding optimization problem is solved with a particle swarm optimization method. Numerical results are presented for the parameter identification of several elliptic and hyperbolic model problems.