Most of the family's Colonial era tombstones that I've run across in Connecticut spell our surname as "Parmele," while "Parmelee" and "Parmley" are most popular today, according to the U.S. Census Bureau.
Few early European immigrants could read or write. Life in the 17th and 18th centuries was harsh. People struggled just to survive, to find food, shelter and protection from Native American tribes and other European settlers. Many relied on others to do their writing for them. Early legal, church and census records often reflected what the listener thought was "right." Spelling wasn't paramount -- getting the message across was.
Recently I discovered North America's immigrant ancestor John Sr. was literate. While his name appeared in 16th and 17th century English parish records with various spellings, his signature can be found on a 1608 counterpart deed of covenant in Lewes -- as witnessed by Wm. Thomas, John Rowe, John Hood and Edw. Newton -- as "John Parmele." Still, his 1659 will and inventory filed in Connecticut gives his last name as "Parmaly" and "Parmely," and son John Jr.'s as "Parmile" -- all in the same document! (And, no, that is not his signature on the contemporary copy of Guilford's Plantation Covenant.)
The Civil War and the invention of the telegraph helped bring uniform spelling to the forefront in the 1860s -- to get the right dots and dashes transmitted to let the folks back home know if that Union or Confederate casualty was their boy. But even as late as 1870, census records show, one in five Americans over 21 was illiterate.
Various branches of the family have handed down stories about why the spelling of their surname was changed -- a dispute between father and son, a run-in with the law, etc. Tales like these are common in most families, and they may very well be true -- or not! Take them with a grain of salt.
Bottom line: There is no "right" way to spell our family name. Don't obsess over it!
However, it's important to note that "Palmer" (and a corruption of it that's come to be "Parmer") and "Parmenter" and similar spellings are not associated with this family.