I assume you mean the Henry who lived in Reading, Berks County, PA and the Annie buried in Charles Evans Cemetery.
The 1870 census has Henry in Reading and the 1880 census shows Annie a widow which means Henry's death is unlikely to show up on any vital record. This basically leaves newspaper death notices, court and church records.
Fortunately there are several excellent sources for doing genealogical research in Reading: Berks County Genealogical Society. Historical Society of Berks County and Reading Family History Center. The Berks County Archives, the repository for county court records, is also very nice source.
The Archives would have any applicable estate records. One way to narrow down when Henry died would be through county tax records. The Berks County Recorder of Deeds would have any property transfers and sometimes the heirs sell any land the deceased had. The record might indicate when he died.
Church records are probably your best bet and Reading had a number of churches. Do you know their religion? If not Annie or Augustus' obits might.
Sometimes death certificates indicate the exact place the parents were born. It's not that often, but it's worth a try. Augustus' obituary might list surviving siblings. In addition to the sources above try the Reading Public Library for his obit.