We construct massively parallel adaptive finite element methods for the solution of hypebolic conservation laws. Spatial discretizatin is performed by a discotinuous galekin finite element method using a basic of piecesise legendre polynomials. Temporal discretization utilizes a rung-kutta method. Dissipative fluxes and projection limiting prevent oscilations near solution discontinuities. the rsulting method is of heigh order and may be parallelized efficiency through computations on a 1024-processor nCUBE12 hypercube . We present results using adaptive p-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 by overlapping neighborhoods of processors that each perform local balancing