No, I think Parker wins by the method of challenging them to a Princess Bride esque drinking contest. All four get a strong drink, and one or more have drugs in it that'll KO whoever drinks it.
The trick? They've all secretly got the drug in it, so trying to trick the others into this mug or that one after trying to figure out which has it, is futile.
The winner? Parker, through a method none of the others would think of:
Don't drink the drink.