On 1851 census there is a John and Judy Tracey either living with or next door to John and Bridget Tracey. Strong possibility that these are his parents?

1881 census Stratford Perth North
www.familysearch.org
SWIFT
Matthew, 35, b Ont, machinist
Ann 30, Ont
Cathleen, 4, Ont
Anna, 2, Ont
Thos Rath, 12, Ont
Bridget Tracy, widow, 46, Ireland, ...weaver.
All Catholic.

1891 census same place, Ann is listed as Anastasia and Bridget is still with them in Stratford.

1911 census www.automatedgenealogy.com same place. They are both alive and Bridget is still with them with immigration date of 1850.

Here is Bridget's death registration but unfortunately no parents are listed.
Bridget Tracey,widow, 89yrs11mos29day, b Ireland, retired, 81 yrs in Canada, 2 yrs at place of death, parents both b Ireland, informant: Mrs Percy Patrick, granddaughter, of Stratford Ont, burial: Old RC Cemetery, Stratford, 4 Jun 1923.
I was hoping the parents names would be here but no luck.

I cannot find a death rego for Ann Swift which may give her mother's maiden name. She must have died between 1911 and 1919 so that gives you a place to start looking.
As you have likely discovered, Irish records are very difficult to research unless you know the parish.
Were there any other children? If so, their marriage or death regos may have Bridget's maiden name on it.
I could not find a death for John Tracy. He is likely buried in the same cemetery.
Sorry, not much help.