The pulsed nature of laser-driven ion sources and their relative large emission angles result in the production of secondary, undesired, pulsed neutron (and photon) radiation. Conventional neutron monitors struggle to accurately measure in such environments, yet characterizing these fields is crucial for applications like hadron therapy. Parasitic neutron dose measurements were performed at the Petawatt beam of the Dresden Laser Acceleration Source (DRACO) employing laser energies from 4.5 to 18 J. An active extended-range neutron REM counter specifically developed for pulsed neutron fields, the LUPIN-II, was employed, as well as a passive extended-range neutron REM counter, the Passive LINUS. Neutron doses were recorded on a single-bunch level with values up to about 260 nSv per proton bunch characterized by a proton cutoff energy of about 60 MeV at about 2 m from the DRACO vacuum chamber, confirming the expected pulsed nature of the neutron field. Results of passive measurements were compared to the LUPIN-II results, integrated over the same period, and showed a reasonable agreement, confirming the presence of pulsed neutron radiation in the proximity of the DRACO ion source. These results demonstrate for the first time that this kind of radiation can be monitored, in terms of H*(10) on a single-shot basis by using the LUPIN-II neutron REM counter.