We describe and examine the performance of adaptive methods for solving hyperbolic systems of conservation laws on massively parallel computers. The differential system is approximated by a discontinuous Galerkin finite element method with a hierarchical Legendre piecewise polynomial basis for the spatial discretization. Fluxes at element boundaries are computed by solving an approximate Riemann problem; a projection limiter is applied to keep the average solution monotone; time discretization is performed by Runge-Kutta integration; and a p-refinement-based error estimate is used as an enrichment indicator. Adaptive order (p-) and mesh (h-) refinement algorithms are presented and demonstrated. Using an element-based dynamic load balancing algorithm called tiling and adaptive p-refinement, parallel efficiencies of over 60\% are achieved on a 1024-processor nCUBE/2 hypercube. We also demonstrate a fast, tree-based parallel partitioning strategy for three-dimensional octree-structured meshes. This method produces partition quality comparable to recursive spectral bisection at a greatly reduced cost.