We study the identification problem for third-order linear time invariant positive systems in experiments where the output is a Poisson process. The problem well-posedness is investigated when the input-output model is described by a sum of real exponentials. A maximum likelihood procedure is then proposed and the admissible set for the unknown parameters is characterized. The novelty of the approach consists in solving a constrained maximum likelihood problem to estimate residues and eigenvalues based on theoretical results on minimality of positive realizations recently obtained in the literature. Numerical results are also provided.