I'm not familiar with the Healey handbrake system, but if it is of the "fly-off" style, ie, push button to lock up, and lift and release the handle to unlock the parking brakes, then it's similiar to my MG. I had the same issue where the handle couldn't be locked up with the button. If you even thought about bumping it, it would release.
Turned out the parking handbrake cables to the rear brake drums were too loose. You may have to remove some carpet, but somewhere in your system, there should be an adjustment for those cables. Tighten them up a couple of turns. You need to find the "sweet spot" where the cables are tightened enough to hold the parking brake handle in the locked position, but when released, the brakes are not dragging. I jacked the rear axle up while adjusting the cables, and rotated the rear wheels by hand to verify the brake shoes were released when the handle was stowed.