We present a wavelet-based adaptive method for computing 3D flows in complex, time-dependent geometries, implemented on massively parallel computers. The incompressible fluid is modeled with an artificial compressibility approach in order to avoid the solution of elliptical problems. No-slip and in/outflow boundary conditions are imposed using volume penalization. The governing equations are discretized on a locally uniform Cartesian grid with centered finite differences, and integrated in time with a Runge--Kutta scheme, both of 4th order. The domain is partitioned into cubic blocks with equidistant grids and, for each block, biorthogonal interpolating wavelets are used as refinement indicators and prediction operators. Thresholding of wavelet coefficients allows to introduce dynamically evolving grids and the adaption strategy tracks the solution in both space and scale. Blocks are distributed among MPI processes and the global topology of the grid is encoded in a tree-like data structure. Analyzing the different physical and numerical parameters allows balancing their individual error contributions and thus ensures optimal convergence while minimizing computational effort. Different validation tests score accuracy and performance of our new open source code, \texttt{WABBIT} (Wavelet Adaptive Block-Based solver for Interactions with Turbulence), on massively parallel computers using fully adaptive grids. Flow simulations of flapping insects demonstrate its applicability to complex, bio-inspired problems.