Carbon River rd used to be open to the public beyond this parking area, but from what I understand, it's now a trail due to road damage back in ~2007. The Carbon River rd trail is used to access various trails along this section of Mt. Rainier National Park like the Rain Forest and Old Mine trails.
There's a National Park fee for parking, so make sure to pay before you hit the trail.
In late Winter and early Spring, you'll likely hear and see waterfalls through the trees - though, I could not find any trails that lead to them. The trail is littered with massive ancient trees, but I forgot to take any decent pictures of them. Regardless, they're definitely worth seeing. Also, plenty of deer to be found along this trail.