In this article, we introduce the joint maximum a posteriori state path and parameter estimator (JME) for continuous-time systems described by stochastic differential equations (SDEs). This estimator can be applied to nonlinear systems with discrete-time (sampled) measurements with a wide range of measurement distributions. We also show that the minimum-energy state path and parameter estimator (MEE) obtains the joint maximum a posteriori noise path, initial conditions, and parameters. These estimators are demonstrated in simulated experiments, in which they are compared to the prediction error method (PEM) using the unscented Kalman filter and smoother. The experiments show that the MEE is biased for the damping parameters of the drift function. Furthermore, for robust estimation in the presence of outliers, the JME attains lower state estimation errors than the PEM.