A parameter identification problem -- PIP -- is meant to search the relationship between unknown parameters present in a state equation and the knowledge of indirect measurement of variables assigned to the solution of such an equation. In general these problems are ill-posed so that their numerical solution involves regularization techniques as well as discretization of the parameter-to-output mapping, that is, the operator that maps the parameter set to its related observed data. This is a computationally expensive procedure, as the solution of the state equation -- for many choices of the parameter set -- must be generated. This paper describes an alternate approach to numerically solving the PIP, namely an iterative method based on the Levenberg-Marquardt sequential quadratic programming (SQP). A Galerkin type discretization on the product space for parameter sets, state variables and related Lagrangian variables is shown to lead to a sequence of well posed indefinite systems. Convergence for the quadratic programming problem which arises at each iteration step is then proven. The overall minimization procedure is also shown to be convergent. Computer experiments that support the algorithm discussed are presented.