Question: When learning to ride, my dad taught me to always have the inside pedal up when cornering. He reinforced the lesson by telling me about a race he'd been in where someone clipped a pedal when cornering and took down 20-30 people in the pack behind him as he went down.

To me it's quite intuitive to keep that inside pedal up - it keeps it away from the ground when leaning into a corner and I'll often even hang my knee on the inside as I'm leaning to lower the center of gravity even further. Additionally, I *think* that I'm actually pushing on the outside pedal as I go around, which also feels very natural to do.

The vast majority of people I see riding these days (recreationally or light competitive, not professionally), including (much to my chagrin) my own children, insist on putting the inside pedal down when turning. I suppose that there is a belief that "my foot is closer to the ground, so if I start to crash I can save myself more quickly", but it just seems so totally counter intuitive and unnatural to me. It feels very unbalanced to have that inside leg down, like I'm going to fall over on it.

Why does this feel like the "right" way of going around a corner to so many people?

Answer: 
I guess it's because when you lean to the right you'd naturally expect your weight to be on your right-side leg -- IOW your right leg is straight and load-bearing, and your left is bent/relaxed i.e. off the ground.

Once your pedal ever scrapes the ground I think you change your mind about the correct way to do it.
Been at races where the competitors have ground down the pedals to allow continued pedalling through the turn. I often keep my cadence through a turn on road and typically will stutter on a trail turn because I lack confidence in the traction holding. There are times when negotiating a tighter turn or when an obstacle or oncoming traffic forces a path change mid-turn that my stroke ceases, my turn side leg is up and that side's knee is leaned away from bike (into the turn) which gives me a tighter response...the turn becomes sharper.  

The benefit of my outward movement of the knee is questioned by some, and I've linked an article by Sheldon Brown that outlines his, and other bicyclists, views. 
It's most critical in turns to keep on top of the saddle and lean with the bike. Perhaps that's better said, keep your body in line with the center of gravity of the bike. Here's a quote from the artcicle I've linked below:
"Leaning the upper body and the bicycle together, keeping them in line as when riding straight.
This technique has the advantage of keeping the steering axis, tire contact patches and center of gravity all in the same plane. This preserves the proper handling characteristics of the bicycle, and makes a skid less likely."
Best to have turnside pedal up. Seems that's what you'll witness an experienced cyclist go to in a hot corner.