It's not just about having a knack for choosing less busy days in Yellowstone National Park. There are rules involved and permits to get if you'd like to do some backcountry camping in the first national park for all to enjoy.
Yellowstone Park technically never closes, but most of the roads do. Beginning November 1 there are just two gates open (weather permitting) throughout the winter.
Government shutdowns are very confusing events, and it's not always exactly clear what happens with things like National Parks. Different things have happened during different government shutdowns over the years. Not all shutdowns are the same. Administrations have important decisions to make about actually locking the gates.