If the hot air balloon is well anchored you can probably just use your original plan. The winds aloft will push the balloon in the direction the wind is going, so the anchor rope should be at an angle relative to the ground. When you drop the parachute, those same winds should keep the parachute/payload pushed clear of the anchor rope. So long as the drag rope connecting the payload to the anchor rope is sufficiently long and the mechanism connecting the drag rope to the anchor rope is sufficiently slippery, you shouldn't have a problem.
I worry about Energizer777's solution though. Mainly because the rope might be angled because of the winds aloft, as stated before. Payloads on a parachute like to hang right below the parachute (more or less), and if the angle on the rope is sufficient the payload will simply end up resting on the rope itself, which might foul up and collapse the parachute since it's load will not be pulling on it properly.
All of that being said, that solution will work if you do this on a day with very calm, or non-existent wind...
unfortunately our "brand" depends on the parachute as a means of deployment
– nothing like letting your brand dictate engineering solutions! This seem backwards to anybody else? $\endgroup$