We camped in Bozeman, Montana the night before and entered at the West entrance on Father's day. The drive into the park was beautiful on its own.
From what I have heard about trips to Yellowstone, Father's day weekend must be an ideal time to arrive. The park was not at all crowded. We've heard stories of long traffic jams throughout the park. Apparently, everyone else was busy honoring their dad on Father's day because we hardly saw any other cars. Sorry, Dad.
However, we did see buffalo almost immediately upon entering the park.
The rivers were really high. This was actually the case for about every river we saw along the drive through Washington, Idaho and into Montana.
There was snow on the ground in the passes, which shouldn't surprise anyone. The passes were over 8600 feet.
When we stayed in the park, we camped at the Fishing Bridge campground. This was a great location requiring hard sided vehicles. We figured out why on our last day in the park when we saw a bear wandering through the campground as we were pulling. It did make me rethink having let the kids play in the snow in the woods near our spot.
The campground is over 7000 feet high, meaning it was very cold at night! We had some very cuddly kids every morning. The kids even played in the snow around the campsite using their sand buckets.