No month, we've been snowed on twice in Yellowstone in different Julys. I80 has been closed by snow every month except one. When you get up in the mountains the weather is unpredictable. There was 8 inches of snow on the ground and more come down fast in Yellowstone when we left at the beginning of this October. And we got snowed on in Bryce several times this last late May
That being said, mid to late June is -usually- snow free in southern Utah. All of this varies from year to year, some are heavy snow years others are not. The higher you get and the further north the more the weather gets unpredictable.