Assimilation of observations into time-evolving ice sheet models is increasingly recognized as a powerful method to infer uncertain model initial conditions that lead to realistic model integrations with little (ideally no) artificial drift, suitable for projections. To date, such assimilation has consisted predominantly of time-independent inversions of surface velocities for basal traction, bed elevation, or ice stiffness, and has relied primarily on simplified glaciological stress balance models. To overcome limitations of such “snapshot” inversions – i.e., their inability to assimilate time-dependent data for the purpose of constraining transient flow states – we demonstrate the concept of using an adjoint of a time-dependent parallel glaciological flow model. The model used for demonstration implements a hybrid shallow shelf-shallow ice stress balance, solves the continuity equation for ice thickness evolution, and can represent the floating, fast-sliding, and frozen bed regimes of a marine ice sheet. The adjoint is generated by a combination of analytic methods and the use of algorithmic differentiation (AD) software, such as the open-source tool OpenAD developed at the Argonne National Lab (ANL). Several experiments were carried out with idealized geometries and synthetic observations, including inversion of time-dependent surface elevations for past thicknesses, and simultaneous retrieval of basal traction and topography from surface data. Flexible generation of the adjoint for a range of independent uncertain variables is exemplified through sensitivity calculations of grounded ice volume to changes in basal melting of floating and basal sliding of grounded ice. The results are encouraging and suggest the feasibility, using real observations, of improved ice sheet state estimation and comprehensive transient sensitivity assessments.