What's So Funny?
This joke has a lot of cultural aspects to it, and a lot of religious vocabulary as well. It relates to stereotypes of Roman Catholic priests, Protestant ministers, and Jewish rabbis.
Priests are often characterized as Irish, and in fact, most of the nation of Ireland is Roman Catholic. Priests are called "Father" by members of the Catholic church. Flannery is a common Irish name. In Catholicism, priests may sprinkle holy water on people to bless them. In the joke, the priest has successfully converted his bear after putting drops of holy water on its head. Catholics mention Mary, the mother of Jesus, more often than Protestants do. The priest's superior, the bishop, will come to perform two Catholic rituals on the bear to finalize its conversion.
In the U.S., Protestant ministers are stereotyped as being loud, enthusiastic about their religion, and having Southern accents. Evangelical ministers preach "fire and brimstone" sermons with the message that if you don't believe in God, you will burn forever in hell after death. Men in the U.S. South are stereotyped as having double names (like Billy Bob, Jimmy Jack, or Bobby Joe). Sometimes the Bible is referred to as "God's Holy Word" or "Scripture." Many evangelical ministers do not believe in merely sprinkling new believers with a few drops of water. Instead, they prefer to immerse their entire body under water in a river or a baptismal pool.
Finally, Jews are stereotyped as having New York City accents. In fact, many Jews do live in NYC. (To hear a much better NY Jewish accent than mine, check out this link.) Like Christians, Jews believe in God, but they don't believe in Jesus. One job of a Jewish rabbi is to remove the foreskin (of the penis) of young boys in a ritual that welcomes them into adult membership of the religion. This is not a good idea to try with an angry bear, however.