Well, checking one source, the Charles M. Taintor book of 1847, your Charles would be listed in the 9th generation of Taintors, IF he is/was included in this book, there are MANY Charles Taintors.
There is only one listed for 1838, none for 1839 and several in the 1840's.
So, the Charles I found for 1838 is: Charles C. Taintor, born June 1838.
His parents are: Jason A. Taintor, born April 1808 in Windsor, CT & married Maria T. Green, born June 1828.Their children are Jason A., Phineas W., Erving, Eli N., & Albert J.This line is directly related to mine & I have the history of Erving (his Bible handed down from his wife's family).
IF this Charles is yours, I can provide more on the line, back to the early 1500's.
It seems that in April of 1887, when your Charles died, that NY would have a record of parents listed on the DEATH record, which would include his mother's MAIDEN name & his father's, filed with the state OR better yet, the county clerk of where your CHarles died in NY.Michigan required this in the 1860's.
Have you tried NY Vital records or the County Clerks office?You might be very surprised at what you find.
Michigan has posted a lot online that are pre 1897 i.e cause of death, age at death & parents names of the deceased & the residence of the deceased & where born, certainly NY kept good records if the wilderness of Michigan did so.