The problem with parm is that "fake parm" can just be literally the exact same product, but just made outside the borders of the legally defined region, or even made within the region with the same methods, but not under the control of "big cheese". It can still be a high quality product.
Counterfeit honey is a big problem. Honey is mostly glucose and fructose, which you can just buy. You can detect a lack of the pollen you'd expect in real honey, but that only makes it so that you can thin out real stuff. There's other methods to detect it, but it's on ongoing arms race. Buy honey from local beekeepers you trust, if you can. P.s., there idea that local honey helps with allergies is bunk because allergies are typically caused by windborne pollen, which bees dont collect.
Maple syrup has similar issues.
Seafood and truffles are commonly "fake", as in substituted with cheaper stuff.
Not "counterfeit", but a similar problem in Mexico is that the cartels have gotten into the avocado industry.