When working with most forms of High Altitude balloons, it is important to have a near foolproof brand, strong yet simple. James Crist from Flying Apple Space Technologies says the following on the subject:
"Parachutes with a few strong shroud lines work best. We have a Fruity Chute with twelve nylon shroud lines, that has only worked once out of the five times it's been sent up on a balloon. By contrast, the 8-foot parachute from Rocketman parachutes (link) has four strong shroud lines, and has worked perfectly every time."
Often, when a light parachute is sent up free and lightly anchored, it becomes tangled and fails to preform properly on the way down. Be careful of using a parachute with superfluous lines, as to forestall any possible complications.
As a general rule, the thinner the lines, the less suited a parachute is for balloon flight.