Hi Jack and glad that you have found and are working on this problem in WikiTree.
I see a great number of problems with this lineage. I don't see anything that documents or shows a probability that the parents of Jack Parker/Parkhurst are the people attached. This profile has him born Cornwall and then you see the parents are supposed to be of Lancashire but somehow married in Cornwall and then returned to Lancashire. There are some notes in the Comment box re Cornwall as place of birth/death for the father that are worth running down. It seems to me the parents should be marked Uncertain and absent any documentation found over a period of time should be detached.
And, I see (once again) someone has cited Richardson Magna Carta (on father profile) with a link no one can access and does no give a full citation as to what Richardson is citing as evidence. Richardson is not a primary source and any citation to him should contain his primary source citations since his work cannot be viewed unless owned (contrary to the belief of some, errors have been found in Richardson's voluminous work). With all of the problems on these profiles, we have no way of knowing if the Richardson citations even apply to this individual.
Doing some searches at Ancestry, Find My Past, Free Reg, etc., I find no primary source documentation of a Joseph Parker or Parkhurst born even approximately 1629 anywhere in Cornwall or Lancashire (I am not suggesting these searchable sites have everything, just saying not found there). Nothing jumps out at me at American Ancestors, either. However, there are some 'hits' in the periodical search at AA that might be worth running down...the list is long but most hits can be discounted by location or time.
This doesn't help with the identification of the parents of the subject profile but I think it shows there's no primary or reliable secondary documentation yet found to establish a familial relationship.