What is the source for the title "Sir"?
There is no information in the available sources that he was knighted:
https://books.google.co.za/books?hl=ru&id=hV8BAAAAQAAJ&q=Pa...
It is not only knights that have a Sir prefix: the profile notes on his son Robert have Sir Cecil Palmer becoming the eighth baronet in 1826. Baronets are simply commoners, but have a Sir title.
There is a strong possibility that Sir John was a baronet, in which case a suffix of Bt or Bart would be appropriate.
As an aside, I note that this tree branch is accumulating suffixes (snr, jnr, I II) within the name fields. Besides this being against Geni conventions, they were never known by these names, but appear to be a thought of individual genealogists.
The trouble with this is that no one kept records prior to James I officially founding the rank in 1611, and there is a great deal of confusion between "baronet" and the older title of "knight banneret". (The latter title, however, was *not* hereditary, as it could only be granted by the sovereign on the field of battle, and the recipient usually was already a knight.)
As to designations of order within the family, putting them in the suffix field beats all billy heck out of the older practice of *making up* a *fictitious* middle name and arbitrarily inserting it. (Middle names are a Tudor, probably post-Dissolution, innovation, and remained *extremely* rare until the 18th century, when they began to "catch on" with the aristocracy, who were then copied by the middle class, and so on until about 1800, by which time more people had them than didn't.. It is worth noting that Byron, a lord, and Shelley, an MP's son, had middle names, while Keats, born into the working class, did not.)
Oh by the way, the note under Robert Palmer is *erroneous*: the "Sir Cecil" who was the "8th baronet" was a Bysshop, not a Palmer.
"Thomas Palmer sold the house [Parham] in 1601 to Sir Thomas Bysshopp, who came from Henfield. For 320 years Bysshopp descendants lived at Parham. In 1826 Sir Cecil, 8th Baronet, became the 12th Lord Zouche," https://www.parhaminsussex.co.uk/the-house/history/
It seems that the Baronet does not refer to this John Palmer, as this title was created in May 1611.
https://www.britannica.com/topic/baronet
There are no sources attached to confirm the date of birth and date of death of this John Palmer.
There is no information on these dates in the available secondary sources:
https://books.google.ru/books?id=hV8BAAAAQAAJ&q=Palmer#v=snippe...
https://archive.org/stream/visitationscoun00owengoog#page/n37/mode/1up
But it is clear that this John Palmer flourished in the second half of the 15th century.
At least his son Sir Edward died in 1517.