With gold, inflation doesn't tell the whole story. In the modern era of FED actions heavily manipulating interest rates, the real metric to look at is real interest rates: (nominal interest rate - inflation rate). Gold does extremely well when there are negative real interest rates, even while the direct correlation to inflation sometimes looks weak due to interest rate moves. It did great in the 70s with very high inflation that dwarfed the high interest rates, and great in the 00's with low inflation but low interest rates. Still, when inflation is very high nothing beats gold in protecting your purchasing power.
To perhaps over-simplify, the PP values gold as an asset for a few reasons:
1) It performs very well in times of negative real rates (including very high inflation).
2) It is generally uncorrelated from both stocks and bonds, reducing volatility in your overall portfolio.
3) In its physical form, it is hard asset insurance against financial and currency failures.
And FWIW, if you'd like to learn more about the Permanent Portfolio, the best web resource by far is crawlingroad.com and the accompanying forum.