I didn’t have any info on your William or Edward so I had to go looking.Your statement that Edward was born about 1856 made me wonder about his father being Jackson Rhead so I started there.The 1880 census for Yonkers shows Jackson and his family, including a son named Edward and another son named William.
On the 1900 census for Yonkers, there is an Edward Read and his family.I believe this Edward had his name spelled wrong and he is really Edward Rhead.He has a son named William who is the right age for your William.
I couldn’t find Edward or William on the 1910 or 1920 census
On the 1930 census for Yonkers, William is shown with his wife, Mary and his two daughters.His father, Edward, is still alive and living with him.(I must make a correction on my site as I thought Edward had died by 1930.)
There isn't too much info on William’s WWI Registration Card but if you want to see it, let me know.I also have an obit for Dorothea White, daughter of William Rhead and Mary.
All in all, I don’t have definitive proof that Edward is the son of Jackson but evidence seems to point in that direction.Happy hunting!