"My brother’s baby was baptized in a Catholic hospital by a Catholic hospital chaplain because the baby was in danger of death. Does this baby have to be baptized again in his parish church by a priest or deacon, now that he has recovered? Does anything else have to be done? Should the parish be contacted?"

For a valid baptism, it is necessary that the one baptizing uses the correct Trinitarian formula and baptizes with water, intending to do what the Church does. If a Catholic hospital chaplain baptized your child in the hospital, the presumption is that the child is validly baptized. This baptism should be recorded by the hospital chaplain in the appropriate hospital and church records. Since it is a valid baptism, the child is not baptized again in your parish by a priest or deacon. It is strongly recommended, though, that the parish be contacted, so that the appropriate ceremonies, not used in the emergency hospital baptism, can be supplied in a parish ceremony. These ceremonies may include, prayers, biblical readings, anointing with chrism, and more.  This, along with a note of the child’s baptism in the hospital, are recorded in the parish baptismal book for future reference.