We have obtained the double-differential incoherent neutron scattering cross section of liquid and solid parahydrogen in various thermodynamic conditions using TOSCA, a time-of-flight, inverse geometry, crystal analyzer spectrometer, operating at the pulsed neutron source ISIS. The measured cross section provides direct experimental access to the self part of the center-of-mass inelastic structure factor of the parahydrogen molecules in the system. Data have been corrected for the experimental effects and then analyzed in the framework of the Young-Koppel model and the Gaussian approximation. The velocity autocorrelation functions and their energy spectra have been obtained from a fitting procedure, making use of the quantum generalized Langevin equation and of model memory functions, and finally compared to the most recent results of both molecular centroid dynamics and self-consistent quantum mode-coupling theory. Some dynamic quantities were also related to simple equilibrium properties and simulated through a standard path integral Monte Carlo code. Results are very interesting but still urge for further developments of theoretical and dynamic simulation approaches, as well as for more extensive experimental efforts.