The design and analysis of diffusive molecular communication systems generally requires knowledge of the environment's physical and chemical properties. Furthermore, prospective applications might rely on the timely detection of changes in the local system parameters. This paper studies the local estimation of channel parameters for diffusive molecular communication when a transmitter releases molecules that are observed by a receiver. The Fisher information matrix of the joint parameter estimation problem is derived so that the Cramer-Rao lower bound on the variance of locally unbiased estimation can be found. The joint estimation problem can be reduced to the estimation of any subset of the channel parameters. Maximum likelihood estimation leads to closed-form solutions for some single-parameter estimation problems and can otherwise be determined numerically. Peak-based estimators are proposed for low-complexity estimation of a single unknown parameter.