The information you will be able to find depends on a number of factors. To find church records (which you need to do as civil registration didn't start until much later) you need to know the family's religion and where they lived (parish if not actual townland not just the county).
If John and his brother Laurence died in Ireland after 1864 there should be death certificates for them but the certificates will not give birthdates or birthplaces and will not list parents' names. If either died outside Ireland (Scotland or Australia for example) there might be more information available.
