So unless I'm missing something, it looks like no good sources on either to prove the parentage?
One very useful thing found through the first link is this citation:
"Author: Jacobus, Donald Lines, The ancestry of Lorenzo Ackley & his wife Emma Arabella Bosworth, N. G. Parke, Woodstock, Vt, 1960, 361 pgs, Heritage Quest, Date Viewed: 16 Aug 2005, Pg 100a Thomas Wilmarth b: Eng ca 1615"
So that's new, less precise info we could use for his birth place and death.
I had found the second link, but it doesn't seem to support the parents through any sources. I went through them earlier and the only ones that give parents are ones with literally no sources beyond other trees (if we're even that lucky).