Check with the York Region Branch of the Ontario Genealogical Society. They have a lot of local material.
There was a recent announcement that the "Newmarket Era" newpaper has either just become available online or will be doing so before the end of the year. There's also the Aurora Banner and there must be one in Richmond Hill (and there is always Markham Village/Town itself).
I'm not sure just where Frederick and Elizabeth lived. I was tracing their son George this year. George married Mary Charlotte Phillips who was a sister of my 2x great grandmother, Margaret Jane Phillips Arnold. George and his wife farmed on 2nd Concession Markham.
Hope this helps