Two "Fly Fisherman" ago the man who invented the klinkhamer explained how he hackled parachutes. It involves the method Jack stated -- turning the eye of the hook down and positioning the fly verticle. instead of half-hitching, he finished (using the whip finish) the fly without wrapping the hackle. Then he turned the hook (no need to worry about thread and materials coming unwound) and started the thread at the base of the parachute post, wrapped the hackle (from top to bottom, and not bottom up) and whip finished the fly around the parachute. The author mentioned that the fly is more durable because the fish's teeth can't catch the quill of the hackle and strip it out.
However, I am to lasy for this and use 8/0 of 10/0 thread and save a lot of room at the eye. I can manage to save room by starting my thread mid-shank and not at the eye. It may only be a wrap or two, but it works for me and I tie a lot of parachutes. i don't know if you do this already, but this is how I tie a parachuted fly: Start at the mid of the shank and wrap back. Then ad the tails. Wrap all the way back to where I am going to place the chute, and tie it in (I don't use a thread dam, but wrap around the base w/ 3-4 wraps). Then, I tie my hackle in straight up w/ the dull side facing me. Wrap the back, and dub the entire fly. Wrap the hackle (counter-clockwise - allos the thread to push the fibers out of the way without being trapped under the thread)and whip finish.
Another thing I just thought of was if you are having trouble securing the hackle. Give 3 wraps of thread over the quill and tug fairly tight. This gives you some playing room without having the hackle unwind.