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We provide an efficient algorithm to compile quantum circuits for fault-tolerant execution. We target surface codes, which form a two-dimensional grid of logical qubits with nearest-neighbor logical operations. Embedding an input circuit's qubits in surface codes can result in long-range two-qubit operations across the grid. We show how to prepare many long-range Bell pairs on qubits connected by edge-disjoint paths of ancillae in constant depth that can be used to perform these long-range operations. This forms one core part of our edge-disjoint path compilation (EDPC) algorithm, by easily performing many parallel long-range Clifford operations in constant depth. It also allows us to establish a connection between surface code compilation and several well-studied edge-disjoint path problems. Similar techniques allow us to perform non-Clifford single-qubit rotations far from magic state distillation factories. In this case, we can easily find the maximum set of paths by a max-flow reduction, which forms the other major part of EDPC. EDPC has the best asymptotic worst-case performance guarantees on the circuit depth for compiling parallel operations when compared to related compilation methods based on SWAP gates and network coding. EDPC also shows a quadratic depth improvement over sequential Pauli-based compilation for parallel rotations requiring magic resources. We implement EDPC and find significantly improved performance for circuits built from parallel controlled-NOT (CNOT) gates, and for circuits that implement the multicontrolled X gate C k NOT.