First, it's not true that a market sector is cheap whenever the forward curve lies above the par curve. In fact, whenever the yield curve is upward sloping, the forward curve will always lie above the par curve. Conversely, when the yield curve is downward sloping, forwards will always lie beneath the par curve. In the example you quoted, Ilmanen chose a day on which the par curve is virtually flat to make a very specific point: forwards can amplify small misplacing.
To understand the rich/cheap signals, let's think through the shape of the yield curve first: the forward curve Ilmanen provided (we'll call it "Curve A") is not particularly smooth. But it can be argued that a sensible forward curve SHOULD be very smooth; after all, market participants can't possible have very differing views regarding short-term interest rate 10-years from today vs 11-years from today. So imagine that there is a "fair value" version of the forward curve that's super smooth (we'll call it "Curve B").
Now let's price bonds with the two curves. Because the forward rates on curve A (the wavy one) in the 12-year sector are particularly high, bond cash flows, when discounted with these high forward rates, will result in lower prices. By contrast, the forward rates on the fair value curve B will be much lower – because of the smoothness constraint, the curve can't swing up like that. Accordingly, the fair value of 12-year bonds, when priced using these lower forward interest rates, will be higher. This is why the 12-year sector can be perceived as cheaper than fair value.
In practice, it's probably easier to build a smooth forward curve and calculate spreads relative to this smooth curve, instead of depending on a wavy forward curve to detect rich/cheap signals.