Hey I had a similar problem with a 1994 GMC Yukon. The reverse lights would go on for half a second when changing from park to reverse, but would stay on the whole time while changing from neutral or drive to reverse. I was able to fix the issue by adjusting the neutral safety switch.
It is a plastic part kind of in the shape of one quarter of a circle and is located underneath the steering column and is attached to it about half way down. You can adjust it by hand. You will be able to move it one way and/or the other and it will make a "clicking" noise each time you move it over left or right. If one way doesn't work try adjusting/pushing it the other way. Also, try to keep track of the starting position/how many clicks you move it over in case you move it in the wrong direction first and have to move it the other way (so that you can move it back to its starting position and start from there moving it in the opposite direction).
Have another person watch the taillights for you as you have the parking brake turned on, try changing into reverse and adjusting the neutral safety switch one way or the other until it works properly.
Hope this helps!
Edit:
Whats up guys so my truck has had this issue for ever (literally) and its about time i fix it. its a 90 gmc c1500 RCSB 4.3L. when i put it in reverse the lights DO NOT turn on. If i want them to come on i have to hold the lever up like putting it back into park and something it sounds like the...
www.gmt400.com
^This post goes more in depth/has some good pictures.