Simpson and Tostevin proposed that the width and shape of exclusive parallel momentum distributions of the A−2 residue in direct two-nucleon knockout reactions carry a measurable sensitivity to the nucleon single-particle configurations and their couplings within the wave functions of exotic nuclei. We report here on the first benchmarks and use of this new spectroscopic tool. Exclusive parallel momentum distributions for states in the neutron-deficient nuclei 22Mg,23Al, and 24Si populated in such direct two-neutron removal reactions were extracted and compared to predictions combining eikonal reaction theory and shell-model calculations. For the well-known 22Mg and 23Al nuclei, measurements and calculations were found to agree, supporting the dependence of the parallel momentum distribution width on the angular momentum composition of the shell-model two-neutron amplitudes. In 24Si , a level at 3439(9) keV, of relevance for the important 23Al(p,γ)24Si astrophysical reaction rate, was confirmed to be the 2+2 state, whereas the 4+1 state, expected to be strongly populated in two-neutron knockout, was not observed. This puzzle is resolved by theoretical considerations of the Thomas-Ehrman shift, which also suggests that a previously reported 3471-keV state in 24Si is, in fact, the (0+2) level with one of the largest experimental mirror-energy shifts ever observed.