In contrast to traditional terascale simulations that have known, fixed data inputs, dynamic data-driven (DDD) applications are characterized by unknown data and informed by dynamic observations. DDD simulations give rise to inverse problems of determining unknown data from sparse observations. The main difficulty is that the optimality system is a boundary value problem in 4D space-time, even though the forward simulation is an initial value problem. We construct special-purpose parallel multigrid algorithms that exploit the spectral structure of the inverse operator. Experiments on problems of localizing airborne contaminant release from sparse observations in a regional atmospheric transport model demonstrate that 17-million-parameter inversion can be effected at a cost of just 18 forward simulations with high parallel efficiency. On 1024 Alphaserver EV68 processors, the turnaround time is just 29 minutes. Moreover, inverse problems with 135 million parameters - corresponding to 139 billion total space-time unknowns - are solved in less than 5 hours on the same number of processors. These results suggest that ultra-high resolution data-driven inversion can be carried out sufficiently rapidly for simulation-based "real-time" hazard assessment.