One theme of the novel is fitting into a time or place. George and Lennie have to keep moving around because they are misfits. While Lennie in particular is not well suited to society because of his disabilities, George also has trouble because he has chosen to be a devoted friend and supporter. Lennie is an innocent man who cannot be fully responsible for his actions. Unfortunately, by the novel's end, he can no longer occupy a useful spot in society. George decides he must take on the role of mercy killer, preempting society's judgment and punishment.
Although there is not an exact parallel, Candy's dog plays somewhat the same role as Lennie. The dog symbolizes uselessness to society, despite an individual bond to one person, in ways similar to Lennie. The difference is that the dog outgrows his usefulness because of age. He parallels Lennie, however, in his innocence because of his species's innate limitations, not a particular handicap.
The dog's death symbolizes mercy. He dies first, by shooting, as Lennie will die later. The dog's killing is foreshadowing of Lennie's killing. Although obviously Steinbeck does not mean that Lennie is an animal, he does convey the idea that Lennie is blameless.