I don't know this one; Washington Co. is not far from Franklin Co. KY where a bunch of Parent/Parrent people lived for some years. I don't know of my bunch being connected with them.
But here's what I could find quickly. Rose and Silvester/Salvient/Salbert Parent were in District 2 in Washington Co. KY in 1860; Glenville (possibly same place as 1860), Washington Co. KY in 1870; then to "Botland District No. 13" in Nelson Co. KY in 1880. The census consistently shows he's from Kentucky, not Indiana (could be wrong of course). In 1880 his parents are shown as being from Kentucky.
Back in 1850, Salbert (seems to be the most consister spelling but I won't dispute that it should be Sylvester if you say so), he's in Washington Co. KY then, too, living with a Weakly (sp?) family. Age 19 b. KY.
A John R. Weakly b. 10-May-1841 died 2-Mar-1928 whose mother was Nancy Curtsinger. Possibly Nancy was Rose Jane's sister?
Rose & Silvester had a son Charles b. abt. 1862, he dies 1935, survivors don't know his father's 1st name just last name Parrent. Survivor states mother's name was Jane Nally.
They also had a son William Thomas, also dies 1935, survivor nettie Lang (Lanz?) remembers Jane Weakly as mother's name and Sylvester Parrent as father's name. Death Certificate also says he was b. Columbia or Columbus Indiana (could be "Columbis").
Columbus IN is in Bartholomew Co.; the Parent/Parrent family that isdistantly related to me has a tie-in with Bartholomew Co. IN. Josiah Parent married Elizabeth Barlow in Washington Co. KY before coming to Indiana, and her father's family ends up in Bartholomew Co. IN. Among the children of theirs that I know about, according to census they had an unknown male child born in the 1825-1830 time frame. I assumed he died young. Maybe not.
Rose & Sylvester's daughter Maggie married a Farnsworth. On her DC it says dad's name was Joe Parent, mother's Rosie Wheatley.
Daughter Martha married a Nally, she d. 1949, her DC says father "S. Parent" mother Jane Curtsinger.
Daughter Sarah married a Rhodes, she d. 1950, her DC says father "S. Parrent" mother Jane Wheatley.
If this man was Sylvester Joseph Parent, or Joseph Sylvester, then he won't likely be Josiah's son. Josiah Parent and Elizabeth BArlow indeed had a son Joseph but he's in California by 1860, and died there 26-Nov-1911 in Bakersfield, and was not known to have a middle name.
Well, I hope this information is of some use to you. If you want those death certificates, let me know.