John Fawkes - Parents

I was Googling for more info before adding these connections to my tree. The Fawkes genealogy I found doesn't show Sir WIlliam and his wife Ellen Harrington having a son named John. Also, the geography is off since William and Ellen were both born and died in Yorkshire, whereas thsi John was born in Warwickshire. They may be related but don't seem to be from the same immediate family, as best I can tell.

https://www.stirnet.com/genie/data/british/ff/fawkes1.php

This is what is shown for the immediate family of William and Ellen:

i. William Fawkes (d before 1575)
m. (c1521) Ellen Harrington (b 1490, d 08.1575)
a. Thomas Fawkes of York (d 1581)
b. Edward Fawkes (b 1532, d 01.1578/9)
m. (c1567) Edith Blake
(1) Guy Fawkes (b 13.04.1570, d 31.01.1605/6, 'Gunpowder Plotter')
m. (1590) Mary Pulleyn
(A) Thomas Fawkes (bpt 06.02.1590/1)
(2) Anne Fawkes (bpt 03.10.1568, bur 14.11.1568)
(3) Anne Fawkes (b 1572)
m. (1599) Henry Kilburn
(4) Elizabeth Fawkes (b 1575)
m. (1594) William Dickinson
c. Edith Fawkes
m. J. Foster
d. daughter
m. H. Ellis

There aren't any notes or sources in the profile of this John and, as I mentioned, he doesn't appear as a child of the couple connected as his parents unless there is other documentation for that.