A "parish" is the area around a church where the worshippers at that church live. A typical sized parish would have at most a few hundred people living in it - for example one village and the surrounding farms.
Usually the cemetery or graveyard would be the land immediately surrounding the church building.
There was not much social mobility at that period in history, and each family would have its own area within the cemetery where all the family members from previous generations had been buried.
There are several places in France called Montfaucon, but because of the reference to the "king's barber" and the "pardon," the OP's quote is likely to refer to a district of Paris, which was infamous as a place where criminals were executed.
Presumably the barber was executed in Montfaucon for some crime, and buried near to the place of execution - quite likely in "unconsecrated ground" that was not under the religious jurisdiction of any church, since the criminals buried there would be unlikely to spend their afterlife in heaven!
However, after being posthumously pardoned by the new king, the barber was now entitled to be buried with the customary religious ceremonies, and the three men therefore exhumed the body and took it to be buried in the parish where his family lived.