We construct massively parallel adaptive finite element methods for the solution of hyperbolic conservation laws. Spatial discretization is performed by a discontinuous Galerkin finite element method using a basis of piecewise Legendre polynomials. Temporal discretization utilizes a Runge-Kutta method. Dissipative fluxes and projection limiting prevent oscillations near solution discontinuities. The resulting method is of high order and may be parallelized efficiently on MIMD computers. We demonstrate parallel efficiency through computations on a 1024-processor nCUBE/2 hypercube. We present results using adaptive -refinement to reduce the computational cost of the method, and tiling, a dynamic, element-based data migration system that maintains global load balance of the adaptive method by overlapping neighborhoods of processors that each perform local balancing.