Isn't that just something like using a series estimator to be somewhat nonparametric?  Looks like zero-degree polynomial is the only thing that climate change skeptics believe in. No trends and everything is stationary under the sun...
True, using sixth-degree polynomials would make me a bit nervous too, but it does not raise obvious red flags for me in the present context. At least the authors are careful enough to throw in the minimum amount of robustness checks so the results do not appear controversial if interpreted right. "Right" means with all caveats splashed across the paper, to avoid their results being spinned by agenda-driven political wackos on both sides of the aisle.
To be fair, economists often do more controversial things and sometimes get away with that. My suggestion to critics: "Try to do better if you can with the data available."
Don't economists (and econometricians) use polynomial distributed lags to deal with the curse of dimensionality in VAR?  high solar activity? There are six sunspots now and there have been as many in ancient times when the climate was so warm that there was winepoduction in England and Greenland was called Vineland. In the face of this fact the whole climate scare appears to me like a huge confederacy of dunces.  Whatever the degree of the polynomial, extrapolation is bad; ie. has no scientific worth.  I don't think you get to be a Columbia professor without being pretty politcally smart, and there's nothing smarter than visibly joining the dominant religion.  If they used a 5th degree polynomial instead, they would have predicted an increase in yield!