The storage capacity of a graph measures the maximum amount of information that can be stored across its vertices, such that the information at any vertex can be recovered from the information stored at its neighborhood. The study of this graph quantity is motivated by applications in distributed storage and by its intimate relations to the index coding problem from the area of network information theory. In the latter, one wishes to minimize the amount of information that has to be transmitted to a collection of receivers, in a way that enables each of them to discover its required data using some prior side information. In this paper, we initiate the study of the Storage Capacity and Index Coding problems from the perspective of parameterized complexity. We prove that the Storage Capacity problem parameterized by the solution size admits a kernelization algorithm producing kernels of linear size. We also provide such a result for the Index Coding problem, in the linear and non-linear settings, where it is parameterized by the dual value of the solution, i.e., the length of the transmission that can be saved using the side information. A key ingredient in the proofs is the crown decomposition technique due to Chor, Fellows, and Juedes (WG 2003, WG 2004). As an application, we significantly extend an algorithmic result of Dau, Skachek, and Chee (IEEE Trans. Inform. Theory, 2014).