The absolute, number one thing to do is get a death certificate for both John P. Tarter and Matilda Jane Calhoun.If you know where they are buried, it should be possible to determine the date of death from the tombstone or cemetery records.http://www.chronography.com/death.htmlhttp://www.chronography.com/death.html will give you more information on how to do this and links to the various state pages.Wisconsin has records from Oct 1907, and some incomplete records for earlier dates.
In the absence of a death certificate, try for an obituary.Even if this does not name the parents, it may name a surviving sibling.The trick, then, is to track the surviving sib and get a death certificate for this individual.This will verify the father - the mother you'll then be able to verify from the birth year of the child.Since Squire T. Tarter married Mary "Polly" Schoolcraft in Sept 1849, it's reasonable to assume that the children born prior are those of Polly Turpin, Squire's 1st wife.
Also, look at the 1870 census for both Kentucky and Indiana to see if you can determine exactly where Squire, etc. went.Check your local library - some have good genealogical sections - and look for indexes of wills, tombstones, etc.Try to locate descendants of other branches of the same family who may have done their own research on these families.
The only one I know anything about is Patrick Green Tarter, born ca 1849, who joined the migration from Pulaski Co., KY to Williamson Co., TX and was the postmaster of Waterloo, TX from 1893-1894.Texas has death records from 1903, so he may be one possibility of an individual you can track if you can't find John P.