I found another compelling document that serves as evidence that Pierre Allary was indeed, Elizabeth Lynch's father.
https://www.newspapers.com/newspage/39342883/
Pierre Allary's obituary the Manitoba Tribune paper (May 15, 1924) mentions that he was survived by his children, one of which is "Mrs. Alfred Lynch":
"At the age of 27 years he married Genevieve Zastre, who predeceased him a few years ago. Fourteen children were born to them, nine of whom are still living. ' They are: Hyaclnthe, Napoleon, Eljear, Joseph, Mrs. Alfred Lynch, Mrs. J, Brown, of St. Eustache: Louis; Mrs. Fraser, of Olga, N.D.; Mrs. Desjarlals, of Togo, Honk. Seventy grand - children, 41) great - grand - children, and four great - great - grand - children survive "le vleux UC Pierre."
In combination with all the earlier answers, I think it's safe to say that Pierre and Genevieve Allary are her parents (until proven otherwise).
Thank you all for the help!