I do not know of anybody who argues that AD and PD are prion diseases.

The various prion diseases (Creutzfeldt–Jakob disease among others) arise when prion protein (PrP) misfolds. The misfolded form of PrP has pathological effects. The misfolded form of PrP can catalyse conversion of the normally-folded form of PrP into its misfolded (pathological) form. This leads to an exponential increase in the level of misfolded PrP.

AD and PD are both protein misfolding diseases. In AD the main suspect is Aβ (there are probably others). Aβ can misfold, which leads to the formation of amyloid - this is why insoluble amyloid plaques are characteristic of AD. The evidence suggests that insoluble amyloid plaques containing Aβ are relatively inert - what does the damage is soluble Aβ complexes along the amyloidogenic pathway though the mechanism(s) of this are controversial. There is no evidence that I know of that the misfolded form of Aβ can catalyse conversion of other molecules of it into the misfolded form, as happens in prion diseases.

PD is similar - there is good evidence that misfolded α-synuclein is neurotoxic. However, I don't know of any evidence that misfolded α-synuclein can catalyse conversion of its normal form into a misfolded form, as in the prion diseases.