I followed the recommendation from the original poster and parked at the Randall's at 38th St. This worked out well, but I didn't care for the several-block stretch on Lamar, which felt unpleasantly exposed and hectic. Once around the corner and down along the creek, though, there were lots of cool, shady tunnels of green, and recent restoration work has added a bunch of nice natives, including bigtooth maples, Mexican plums, and a variety of oaks. I'll definitely come back here, especially in the summer.
The downsides are that it's a pretty busy trails--lots of joggers and cyclists--and the absence of bathrooms (that I could find, anyway). Next time, I'll start further south, probably in the parking area off Shoal Creek Blvd.