Quite a few answers given herein are just plain wrong. Some answers were technically correct but failed to answer the question as to what keeps the chutes apart (it's not "in the rigging," though my hat's off to riggers).
"Air spillage" is the technically correct answer, and some of the loads we spit out the back had eight chutes. I've seen heavier systems for C-17s with up to twelve chutes. It still works.
Symmetrical chutes will separate, so asymmetrical design isn't the answer.
Some chutes have vents, some don't. But aside from JPADS, vents for cargo chutes are symmetrical.
The parachute cords are not of differing lengths.
Cargo chutes for Apollo most certainly had vents.
Cargo chutes for JPADS most certainly have vents (gores) and are computer-steered to precision landing at the PI, even with 60k lb loads.
Personal found chutes often have vents to aid in stability and provide some forward motion. They're normally closed at opening, and the parachutist performs a "four-line modification" which simply releases the panels and allows him/her to steer the chute.
MacLaddy: "So the air being released from each chute on the sides is forcing them apart? Sort of a spill-over effect." Bingo! It's that simple. Kudos for the new guy!
Someone asked about the length of lines for cargo chutes - yes, they're longer, proportionally, to canopy size as compared to personnel chutes. This reduces the deflection angle required for complete separation.
"Tension lines" aren't the answer.
Borek mentioned forces acting on the parachute are all pulling them towards the center. Mostly true. However, with the spillage keeping them apart, they're canted, and will "fly" a bit the same as a towed parachutist (training and entertainment) behind a boat or vehicle rises when towed, even with a plain vanilla round T-10C. However, this only lessens the centering force, making it easier for spillage to help keep them apart.
Sophiecentaur is correct in stating that the air traveling past the periphery is moving more rapidly than the stagnation air spilling out from under the chutes which rises between them. However, the momentum change is precisely that required to counter the weight of an airplane or slow the descent of a parachuted object. Wings are typically designed to work rightside up and do a better job of imparting that change in momentum. Some acrobatic wings are largely symmetrical, however, and work nearly as well upside down as rightside up.
Well, that's all, folks. Hope this clears up some things.