Before we can figure out who Jacob's parents were, we need to know who Jacob was.
How do you know Elmer's father is Jacob? Have you seen a death certificate, or Elmer with his parents in earlier censuses/records? Do you know any part of Elmer's mother's name?
I found Elmer in 1900 Toledo, Lucas Co OH as you said, (born Oct 1859 OH-NJ-MI) with wife Lillian and son Thomas among other children. I also see Elmer of Toledo OH (born 1862 OH) married Lillian Hardinger in Monroe Co Michigan in 1885 (no parents listed on certificate).(Lillian is found in 1880 Monroe Co MI with her parents George and Mary Hardinger.) In 1910 in Washtenaw Co MI, Elmer is born 1859 OH-NJ-MI, with wife Lillian & children. In 1920 in Shiawassee Co MI, Elmer is born 1856 OH-OH-OH. In 1930 Washtenaw Co MI, Elmer is born 1862 OH-NY-NY. Elmer's Washtenaw Co MI grave has been submitted to Find-a-Grave (no photo) with dates 1859-1933 and also wife Lillian's with her maiden name reported as "Smith".
But the only Elmer Applegate in Ohio I can find remotely in his age group in 1880 is "Elmore" born 1867 OH, son of Paul & Elizabeth both born in Ohio (also found in 1870). Other Elmers about his age are born in different states and don't live in Ohio. Nothing better in 1860 or 1870 either.
There is a Jacob Applegate living in Lucas Co Ohio between 1850 and 1877, but he doesn't seem to have a son Elmer. This Jacob is born in NJ and his second wife Martha (m. 1857) is reported as born either in NJ (1860 census), MI (1870 census) or NY (death record). Jacob & Martha's children were Jacob (b. abt 1862 OH), John (b. abt 1867 OH) and possibly Eva (b. abt 1876 OH).
I have more information on this Jacob, his family & his ancestry, if you find a connection.