An ancestry.com public tree shows Edward Gerlach born 12 Aug 1862 from Prussia married to Phebe Ann Honeywell born in Iowa on 05 SEP 1867. They had a daughter named Winifred Elizabeth "Winnie" Gerlach, born on 26 Dec 1893 in Williamstown, Chickasaw, Iowa. Edward's parents' names are not known, however. There are several public family trees that list the same information...no information at all on parents of Edward.
There are several August Gerlach's mentioned in the public family trees, but none meeting all your information. One was born 21 Jan
1830 in Feuerbach, Germany and had a son named William E. Gerlach, but William E. was born in Nebraska in 1878. Another August born in Prussia with a wife Wilhelmina Aust but no child with the name Edward or version of it. Neither do the other August Gerlach names show a child named Edward.
Family Search: https://familysearch.org/pal:/MM9.2.1/MWMF-GTZ shows:
August Ferdinand GERLACH born 25 March 1849 in Heiligenbeil, East Prussia, spouse Wilhelmine Caroline Scharfschwerdt, born 2 August 1847, married 25 Nov 1873. They had a son but not an Edward.
They also show a Carl Freidrich August Gerlach born about 1833 married to Henrietta Wilhelmine Vetter and two female children, no Edward.
Shows a Heinrich August Gerlach (1840-1912) married to Lina Wilhelmine Louise Hirsch (born 1848). Again no mention of a child with the name Edward.
Such are the joys of genealogy! Hope you can get it narrowed down.