We have implemented the first-order adjoint sensitivity analysis procedure (ASAP) into the three-dimensional parallel radiation transport code system Denovo, a module of the SCALE software suite. In particular, we used a Krylov-based approach to compute the solution to the inhomogeneous adjoint systems occurring in the ASAP. Our implementation, as a component of Denovo's scalable framework, should allow the efficient computation of cross section and atomic number density sensitivity coefficients for critical systems in a massively parallel fashion. We have constructed a proof that the Krylov-based approach converges to a unique solution and compared its computational requirements with the standard algorithm used in the neutron transport community. In addition, we performed a verification of our ASAP implementation on the Godiva experimental benchmark. We found the new approach to be an order of magnitude faster than the standard algorithm in this benchmark.