There were lots of government agencies shutdown besides just parks, DoD on minimal staff, Social Security closed (so no processing new applications or pushing buttons to make current payments process) It was bad across the board.
Imagine you work for DoD supporting servicemen in overseas area to ensure they get the equipment and supplies they need and being told, that you can't work, you cannot volunteer to come in to help those servicemen, don't know when you will be allowed to come in, and oh yeah, we don't know if you will get paid for this week you are off (no you can't use your leave) and don't know when you can come back. Oh yeah, for those that did work (because you are essential) you go in each day not knowing when you will get paid for your time.. (because payroll people aren't essential) and you still have expenses to get to work.. So yes, Captain Harry people did get paid, but there was a lot of stress on lots of people who didn't have anything to do with the situation either.