We present an adaptive grid, massively parallel 3D implementation of a fourth-order immersed interface method for the incompressible Navier–Stokes equations. The solver maintains fourth-order accuracy everywhere in the computational domain for both velocity and pressure, including near immersed bodies and across resolution jumps. Both stationary and moving boundaries are supported on a background block-structured, multilevel Cartesian grid organized using an octree data structure. The solver integrates several advances from earlier work, including a fourth-order 2D Navier–Stokes immersed interface discretization, a high-order 3D wavelet-based adaptive grid framework, a 3D immersed interface implementation for advection–diffusion problems, and a recently developed high-order 3D elliptic solver with immersed boundaries. To the best of our knowledge, this work represents the first fully high-order 3D incompressible Navier–Stokes solver that supports both stationary and moving embedded boundaries on adaptive grids. In this presentation, we show verification and validation through convergence tests and comparisons with established benchmark solutions. We further demonstrate the accuracy and efficiency of the solver through comparisons with existing results obtained using lower-order discretization schemes, and present parallel performance on multi-node architectures.