Sorry, nothing about Frank OSTIGUY's parents. I'm including the following additional information I have acquired about Frank in case it might be of help:
Another respondent sent the following Canadian birth and marriage references which she believes refer to him:
Birth: Francois Alexis OSTIGUI dit DOMINIQUE, 7 Apr 1800, St-Hilaire, Rouville, Province Quebec, Canada
Marriage: Francois OSTILLY dit DOMINIQUE to Marie Emelie GOGUET, 18 Apr 1825 in St-Cesaire, Rouville, Province Quebec, Canada
Based on US Federal and KS State Censuses: Frank and Emily probably moved to VT between abt 1841 when their 4th child (that I know about) was born in Canada, and May 1843 when dau Elizabeth was born in Burlington, Chittenden Co, VT. Next child Edwin also probably born in VT about 1846. Last known child born in New York about 1847; the family was in Hague, Warren Co, NY in 1850 and 1860. Found Frank and Emily in Shannon Township, Atchison Co, KS in 1865 with grdau Elenor CADIEUX (child of Elizabeth from a first marriage). Elizabeth CADIEUX md. Leander HILL in Atchison Co, KS in 1863.
Frank apparently began using the anglicized Francis or Frank or often just F. S. Austin when he immigrated to U.S. The only time I found OSTIGUY after that was in a 1925 Iowa State Census where dau Elizabeth HILL's father is shown as Frank OSTIGEE.
Emily probably died between 1865 and 1870 because in 1870 Frank was in Vermillion Twp, Marshall Co, KS with wife Eleanor, b. in Ireland abt. 1820. In 1875 the couple was in Wells Twp, Marshall Co, KS. Have not located any info after that.
The respondent mentioned above also sent me the following information about Frank's first wife Emily:
Emily GOGUET's parents were probably: Joseph GOGUET, b. abt 1780 in St-Mathieu, Beloeil, Verchères, Quebec Province, Canada Marie COLIN dit LaLIBERTIE b. abt 1779 in St Francois-Xavier, Verchères, Province Quebec, Canada. Joseph and Marie were md 15 Oct 1798 in Chambly (St Joseph), Montreal (Basilique Notre-Dame),Province Quebec, Canada Emily's birth: 4 May 1804 in St-Hilaire, Rouville, Province Quebec, Canada