You need Annie's marriage certificate to confirm her Father's name and occupation
The certificate may not help if the marriage of Henry Mitchell and Margaret James in Merthyr district in 1877 is the right family.

Modified - Annie's birth name was Hedley
1881 Cwmdu House, Aberdare
Thomas James Head Widower age 57 born Aberdare, Glamorgan Ag Lab
Henry Mitchell Son-In-Law age 26 born Neston, Cheshire Ag Lab
Margaret Mitchell Dau age 28 born Aberdare, Glamorgan
Henry James Mitchell Son age 1 born Aberdare, Glamorgan, Wales
Annie Jane Hedley Step Dau age 10 born Aberdare, Glamorgan Scholar
RG11 Piece 5324 / 87 Pg 6
Birth of Ann Jane Hedlay in 1870 (Sep Q)
Merthyr volume 11a page 429