We propose a parallel shared-memory schema to cooperatively optimize the solution of a Capacitated Vehicle Routing Problem instance with minimal synchronization effort and without the need for an explicit decomposition. To this end, we design FILO2$^x$ as a single-trajectory parallel adaptation of the FILO2 algorithm originally proposed for extremely large-scale instances and described in Accorsi and Vigo (2024). Using the locality of the FILO2 optimization applications, in FILO2$^x$ several possibly unrelated solution areas are concurrently asynchronously optimized. The overall search trajectory emerges as an iteration-based parallelism obtained by the simultaneous optimization of the same underlying solution performed by several solvers. Despite the high efficiency exhibited by the single-threaded FILO2 algorithm, the computational results show that, by better exploiting the available computing resources, FILO2$^x$ can greatly enhance the resolution time compared to the original approach, still maintaining a similar final solution quality for instances ranging from hundreds to hundreds of thousands customers.