I have a question, a suggestion, and a initial guess for you based on your reply.
My question.In what cemetery did you find Julia?
My suggestion.Have you tried obtaining a death certificate for Julia?A death in 1913 should be officially documented.Death certificates will normally name the parents.
A quick scan of my 1850 Ohio Federal Census records shows one possibility.There is a J. Minerva Hungerford, born about 1843, daughter of Henry and Sally Hungerford.They resided, in 1850, in Johnston, Trumbull county, Ohio.Your Julia Emma potentially could be Julia "M." Hungerford.The "M" for Minerva being abbreviated to "Emma."
This is strickly a first guess.I would try to obtain the death certificate.If you do let me know what you find...