Worked at YNP 3 summers near the north Gardiner entrance at Mammoth area. Any day of the year can be long sleaves or a dusting of snow because most of the Park is around 8000 ft. I don't think I would go much before June unless you like jackets. Unfortunately that's too late to beat school getting out. We worked into Oct. and never had snow at our campsite in Gardiner but that's lower than most of the Park. Also, some places will close for the season during Oct. We've found recently that a lot of people go during Fall. I would just go when you prefer, plan, reserve and don't worry about crowds. Maybe you know there is only 1 campground within the Park with any hook-ups; Fishing Bridge(expensive and very busy) but some are so great it's worth dry camping. Gardiner has 2 private campgrounds, West Yellowstone and Cody also. All expensive but with hook-ups.