Today I found out that I had been doing the squat wrong, and I looked into your site for reference on how to do it correctly. All the sources that I had previously looked up said that I should keep my knees behind my toes and squat down with the weight on my heels. However, when I read your squat analysis, it said, "Contrary to propaganda, prominent weight training authorities demonstrate the squat with the knees flexing forward at the same distance as the hips flex backwards" and "If the knee does not travel forward during the barbell squat, the quadriceps muscles are not significantly exercised." I had been told that squatting with the knees protruding out in front of the toes could put too much pressure on the knees and end up damaging them. To make sure that I'm not misunderstanding the article, am I supposed to keep my knees behind my toes (therefore having more weight on the heel of my foot) or have them travel more forward (therefore putting more weight on the ball of my foot)? Also, some people told me to keep my feet parallel during the exercise, but I couldn't find anything on it in the barbell squat instructions page. Does it matter?
You read it correctly. If your knees are healthy, your knee can safely travel slightly in front of your toe provided you warmup, progress systematically and allow for adequate recovery between workouts. This technique allows for a more balanced distribution of torque forces between your knee, hip, back, and ankle.
If your goal is to handle as much weight as possible, such as a powerlifter and you are less concerned about knee extension strength or quadriceps development, you can consider the powerlifting style squat which targets the glutes and hip adductors. With the powerlifting style squat, depending on your biomechanics, the knee may not need to travel as far forward as it does with the bodybuilding style squat.
Perhaps, in opposition from what you have been taught, with either form, keep your center of gravity between your forefoot and heel, distributed equally through the entire foot.
You can keep your feet parallel, if you prefer a relatively narrow stance. I would recommend your feet angle out a bit as shown in the diagram on that same page, particularly if you have a slightly wider stance. Just keep your knees orientated the same directions throughout the movement.
As you can see there are several options. Your form should ultimately reflect your personal training goals and individual biomechanics.
When experimenting with altered form, it is safest to begin with a relatively lighter weight on your first workout, then increase your resistance progressively each workout, until your workout weight is determined. This will permit adequate adaptation and will allow you to maintain better and more consistent form. Also see Adaptation Criteria.