Hello, Edward. While John is not mentioned in the Virginia Gleanings in England article linked from JSTOR at the profile of Thomas Parker, a son John is clearly named in the will of his father (image available with Ancestry subscription here).
However, the profile of John has statements claiming DNA testing proves there is no familial relationship. A DNA specialist would have to review the testing as well as who was tested to know if this statement is supported by the testing. I would urge prudent caution in accepting statements based on DNA evidence for something this many generations back.