Accony School in Kilgeever Parish has a website at www.acconyschool.com with a listing of 3,400+ Kilgeever Parish baptisms (at Louisburgh and Killeen Roman Catholic Churches and Clare Island) from 1850 through 1922 (with some gaps in any given year and a 12-year gap from 1882-1893).
Your Richard Bradshaw and Mary (Joyce) Bradshaw have a daughter Mary Bradshaw listed there; baptized on 10 August 1855 with John Winter and Margaret Bradshaw as sponsors. The community of Lecarrow is listed as their place of residence.
Only two other Bradshaw's listed: 1-Ellen Bradshaw born on 28 June 1852 to Thomas Bradshaw and Winifred (Winter) Bradshaw with Michael Grady and Mary Malley as sponsors and "Glen" as the area of residence and 2-Johanna Bradshaw born on 10 September 1852 to William Bradshaw and Mary Bradshaw with Francis Winter and Julia Malley as sponsors and Gurtheen as the area of residence.
Having copied these records from the photocopies of Latter Day Saints Library's microfilm, I can attest to the fact that the pages of these church records for the earliest records and those prior to 1850 were very difficult to read. The first several decades were designated as a "transcription" of the original records. Therefore, finding your Richard's parents' names and the offspring of Richard and Mary would be difficult-to-impossible to obtain for those years before 1850.
For a fee, the South Mayo Family Research Centre might be able to do more research through their website at www.mayo.irishroots.net .
Good luck.
