Qualification: I am a master parachute rigger, taught the sport for 10 years and have packed a couple of thousand emergency parachutes.
I can put an apartment-size piece of nylon into a purse in under 5 minutes, and guarantee that it will open. How nicely it opens is a different question. It usually takes about 45 minutes to service an emergency parachute - 20 minutes to inspect it, 10 to fold it up nicely, another 10 to stuff it into the pack, then the paperwork. Double that time if it's a large 2-passenger parachute.
The "fold it up nicely" part is to make it go into the pack without any unsightly bulges, and the emergency packs are never a nice open bag like the main - the canopy has to be shaped around the closing apparatus and the spring-loaded extraction chute has to fit in the middle. Very little of this process has anything to do with ensuring it opens when needed.
A packer would have to be spectacularly incompetent to fold a canopy in a way that would cause it to malfunction. If you feel the fabric used to make main parachutes you will find that it is extremely slippery - good luck keeping that stuff in the proper place when it goes into the bag, but you don't need to. These things sort themselves out quite nicely most of the time.
There are two important tools used in packing emergency packs - one is appropriately called the Death Strap. It is wrapped around the middle of the canopy as you are folding it into the bag, and if you forget to remove it the canopy will not open. The other is a locking pin used to close each part of the pack in stages. Your mistake will probably make the front page of the local newspaper and involve a lot of time with your lawyer, so packers typically only own one of each and all tools are counted after each job.
Going back to the incompetent packer for a moment - the main parachute is usually packed by the owner, and very occasionally you need to open and re-close the pack. If you are lazy or rushed it is very easy to close the main pack in a way that prevents it from opening. Owners (almost) never open their emergency packs, and riggers follow a standard process to prevent these errors from happening.