Sulphur Creek is more or less an off-the-beaten path trail. It's never been maintained by the National Parks Service, though it's clearly inside Capitol Reef, and in the past there hasn't even been a map or anything to guide hikers. However, it has been gaining popularity over the past few years and now they have dedicated pull-outs for parking cars at either end of the trail (and although its still an unmaintained trail (on account of...it's a river) you can now get a map from the visitor center).
We didn't know any of that, however, because Andrew and I hadn't been to Capitol Reef since 2012 (just a few weeks before Benjamin was born) and Karen hadn't been to Capitol Reef since 2013 (when Dorothy passed away on Brimhall Double Bridge Trail (though Reid went just a few weeks ago with a scout troop)), so we just started doing what we'd normally done and had Karen drop us off at the Castle Rock trail head. But there's a new pull-out/parking area about a third of a mile before this now (just so you know).
While I nursed the baby, Grandpa took the other kids across the road to the dry creek bed (come to think of it, that's probably why they have the new pull-out area—so that you don't have to drag all your kids across the road to get to the proper trailhead). By the time I caught up with them Benjamin had found a rock that reminded him of a guitar and he was rocking out (in every way possible).