I don't agree. The strength of some parodies or hoaxes is that you use the companies name, this is important to make people think (I believe). Under your definition groups like "The Yes Men" would not be considered parody. http://en.wikipedia.org/wiki/The_Yes_Men
I don't want to go into the example you brought up with Foster Farms, you should realize that this comparison is pretty much flawed.
It is not required to disguise yourself as another to make people think.
Parodies in the past did not depend on fooling people as to the source of words. It is enough to conjure up a thought of the parties parodied, through use of similar symbols/words/methods. One doesn't have to actually pretend to be them in an exact way for parody to hit its mark. Nor in the past have I seen great parodies do this.
I don't want to go into the example you brought up with Foster Farms, you should realize that this comparison is pretty much flawed.