OK, I did a little more looking and found that William Applegate in 1900, New Orleans, had a wife (on next page and I didn't see her)named Rebecca.In fact, both William and Rebecca are still living in 1910. In 1880 William is again living in the Whitlow household in New Orleans, but Rebecca is not listed. Perhaps an oversight by the enumerator. His occupation is ship inspector.
So I looked in 1870 and found William Applegate, age 44, b. KY, coal miner in Bells Mines, Crittenden Co., KY, with wife Rebecca 36, b. KY and a Charles Applegate, age 19, b. IN.Is this your Charles?
I don't know when the state of LA began registering deaths, but if the state doesn't have William's record, the parish of Orleans might and it might have his parents' names.