The Park actually drains into the The Missouri, the Yellowstone, the Snake, and the Wind/Bighorn River (which ultimately joins the Yellowstone in Eastern Montana, which joins the Missouri near the MT and ND border).
The Upper Green River Basin got a bunch or rain, but thankfully did not get the really heavy rain on top of the already high snowmelt. The Green went from 4,000 to 8,000 CFS in a couple days. The days before the rain were the first warm weather after a cold wet spring, and rivers and streams were cranking before it rained. Fontenelle Reservoir (above Flaming Gorge) has been coming up over a foot a day. So there is some water coming, but it is a pittance compared to what is needed. After 80 degrees over the weekend, it was 26 last night, so the snowmelt and streamflows are already dropping...