The preferential attachment (PA) model is a popular way of modeling dynamic social networks, such as collaboration networks. Assuming that the PA function takes a parametric form, we propose and study the maximum likelihood estimator of the parameter. Using a supercritical continuous-time branching process framework, we prove the almost sure consistency and asymptotic normality of this estimator. We also provide an estimator that only depends on the final snapshot of the network and prove its consistency, and its asymptotic normality under general conditions. We compare the performance of the estimators to a nonparametric estimator in a small simulation study.