The book "Walter Palmer of Charlestown and Rehoboth, Massachusetts and Stonington, Connecticut : a 400-year (1585-1985) family history" (1986) has a well researched preface to chapter 1 as well as an addendum (called "Did Walter Palmer have a coat of arms?") that are worth reading. They agreed with Col. Charles Edward Banks' 1930's conclusion that Walter was likely a rural dairy farmer, after rejecting claims that he was the son of a Sir Thomas Palmer. They also suggest further research for those interested.
There appears to there have been quite a few different sets of parents attached to Walter through the decades. This book is refuting a different set of parents from what is on WikiTree or what the Walter Palmer Society suggests is probable.
The book can be borrowed here:
NYGBR also has a pedigree that shows uncertain origins.
There was another g2g on him regarding his wives (which, btw, are covered in another addendum). Perhaps someone else has something new.