Parallel memory-efficient adaptive mesh refinement on structured triangular meshes with billions of grid cells. We present sam(oa) 2 , a software package for a dynamically adaptive, parallel solution of 2D partial differential equations on triangular grids created via newest vertex bisection. An element order imposed by the Sierpinski space-filling curve provides an algorithm for grid generation, refinement, and traversal that is inherently memory efficient. Based purely on stack and stream data structures, it completely avoids random memory access. Using an element-oriented data view suitable for local operators, concrete simulation scenarios are implemented based on control loops and event hooks, which hide the complexity of the underlying traversal scheme. Two case studies are presented: two-phase flow in heterogeneous porous media and tsunami wave propagation, demonstrated on the Tohoku tsunami 2011 in Japan. sam(oa) 2 features hybrid MPI+OpenMP parallelization based on the Sierpinski order induced on the elements. Sections defined by contiguous grid cells define atomic tasks for OpenMP work sharing and stealing, as well as for migration of grid cells between MPI processes. Using optimized communication and load balancing algorithms, sam(oa) 2 achieves 88% strong scaling efficiency from 16 to 512 cores and 92% efficiency in a weak scaling test on 8,192 cores with 10 billion elements – all tests including adaptive mesh refinement and load balancing in each time step.