A sparrow is a bird that is referred to in the Gospels by Jesus as a symbol of a life form that is worthless in comparison to a human. Jesus deliberately points out the sparrows in Chapter Six of the Gospel of Matthew as a lesson in how God cares for all of his creation. If God cares for such an insignificant bird as the sparrow, how much more will he care for man? The sparrow is therefore an example of a bird that is very tiny and unimportant in God's creation. It is particularly relevant that Bede compares a man to a sparrow in order to emphasise how unimportant man is and how brief his life is.
A good overview of sparrows (and many other birds) mentioned in the Bible can be found here: http://www.studylight.org/dic/hbd/view.cgi?number=T946
Here are some relevant passages from this source, with my comments, in italics, about the possible appropriateness of the Biblical references to the story told by Bede:
Psalms 84:3 refers to the nesting habits of the sparrow at the altars of the Lord. The altars of the Lord may be a general reference to various structures in the Temple area which would be attractive to small, nesting birds.
Perhaps Bede's story mentions sparrows because they were associated with closeness to God and to the worship of God.
Two passages in the New Testament refer to the sparrow (Matthew 10:29-31; Luke 12:6-7). In these parallel passages Jesus taught His disciples to have confidence in God's love. The God who cares for all of His creation, even the insignificant sparrow, certainly cares for people.
These passages seem highly significant to Bede's story. Bede's story presents sparrows as alone in a cold, dark, hostile universe except for the brief time they spend in the warm, illuminated house. Bede's story implies that God provides an alternative to the frightening, dangerous experiences of the sparrows after they fly out of the comfortable, lighted room.
I believe that Bede, who was very well-known as a man of faith in his time. He was considered a great historian and "the earliest important prose writer." He was a monk; in his day, only members of the church were educated; they also were responsible (for this reason) for keeping records. He is considered a "man of great scholarship and learning." As a member of the clergy, he would have known his Bible well. The reference to the sparrow, I think, would have been made purposely, as in Matthew, Jesus notes that if God knows the circumstances of each sparrow at every moment, how much more does he watch over man?
Sparrows are also industrious yet gentle birds—often playful. Personally, I can see that a connection could be made between sparrows and humans in that many people are very much like sparrows—especially in Bede's time—hardworking, but child-like and innocent in some ways.
Source: Adventures in English Literature, Harcourt Brace Jovanovich Publishers: Orlando, 1985.
I would go mostly with the former. I think that Bede took the idea that a sparrow is a very insignificant and vulnerable kind of bird. He used that to emphasize the innate weakness of human beings. The fact that the Bible uses sparrows in this way when talking about the idea that God notices when something that insignificant dies would have helped to make that connection in people's minds.