I have to assume the parents were married in Ireland and started their family there, but were forced to move to England during the 1845-1850 famine.
I research the Irish Family History Foundation's database. It contains 12,000,000 surviving rcds for abt 65% of Irish counties. Rcds are increasingly sparse B4 1840, rare B4 1800.
Note 1 - Without a KNOWN parish, the chance of an ancestor's rcds NOT being in this database is AT LEAST 35%.
Note 2 - Only 1/2 of Co. Cork's rcds are in this database.
Note 3 - The RC church kept control over most parish rcds from Tipperary (north). These are ONLY available thru the Tipperary Family History Research center at a substantial fee (regardless of outcome).
For Mary O'Malley B 1831 +/- 2 yrs somewhere in Ireland with a father named Pat(rick), I found 3 rcds: 1830 MAY, 1831 ROS, and 1832 GAL.
For Bri(dget) O'Malley B 1833 ditto, I found 3 rcds: 1833 MAY, and 2-1835 GAL.
For James O'Malley B 1835 ditto, I found 2 rcds: 1833 CRK, and 1835 KIL.
For John O'Malley B 1838 +/- 2 yrs ditto, I found 3 rcds: 1836 MEA, 1838 CLR, and 1839 LIM.
This is a mixed bag of results - there isn't a common county for all B rcds. The family may have moved thus disrupting the pattern, or these may all be false positives. Of 11,167 O'Malley B rcds in Ireland 7,734 are in Co. Mayo and 1,570 in Galway. This is a TIGHT distribution; most likely your ancestors origins lie in these two counties.
I didn't search for Pat(rick) O'Malley as only 1 parish in these 2 counties was keeping B rcds B4 1800.
For Cath(erine), you don't have a maiden name. You can search for her marriage rcd only after you validate rcds for the birth of two or more children.
Suggest you return to the English census rcds and/or other sources to find more precise DOB for Mary and Bridget.
Google the Foundation's name to lead you to this database. Register, opt to be notified as rcds are added and browse. Pls post your search results to my genforum msg.