Over the years, Park made at least two different threaded headset wrenches. The chromed steel 32 mm headset/15 mm pedal wrench (HCW-6) was indeed not well made nor a precise fit.

The far better quality "laser cut" HCW-15 is a 32 mm/36 mm double end (1" and 1-1/8" headset) wrench that is a precise and properly octagonal and works very well on those headset locknut and upper race flats.