Many people believe that the dog is a parallel to Candy.  Both are considered old and useless. However, the connection between the dog and Lennie seems more apparent to me.  The strongest example of which is the death of each. Does the dog's death act as a forshadow of Lennie's iminent demise?  Both the dog and Lennie are loyal to  their leader.  The dog to Candy, and Lennie to  George.  George can not allow Lennie to die the same way Candy's dog did: at the hands of a stranger.  He chooses to take matters into his own hands. Crooks also references Lennie's possible futre without George earlier in the novella.  Crooks scares Lennie into believing that without George, Lennie would be chained up/caged like a dog.  Again, this is something that George eventually realizes he can not allow to happen. 

I have a bit more to add to the previous answer.  Candy believes he is becoming useless, just as his old dog is no longer the functional being he once was.  Candy fears that once he is no longer of use to the ranch, he will be tossed aside too. 

Candy's worry is one that all of the men who earn their living manually fear, even George and Lennie, who dream of the Utopian paradise that will save them from similar fates. 

Candy allows his dog to be euthanized to "put it out of its misery" (Chapter 3), and he later laments the fact that he did not do it himself, letting someone the dog did not know well do it instead.  This parallels the ending of the story, when George himself takes Lennie's life to spare him from having to experience suffering that would have been beyond his ability to understand.  The two situations explore the questions of responsibility, mercy killing, and friendship, which are central to the novel.

In Of Mice and Men, what are some similarities between Candy and his dog and George and Lennie?

In Of Mice and Men, similarities between Candy and his dog and George and Lennie include dedication and responsibility.

Candy feels dedicated to his dog. Even though he is old, feeble, and smells, Candy keeps him around. After George comments on the age of Candy’s dog, Candy replies, “I had ‘im ever since he was a pup. God, he was a good sheepdog when he was younger.”

George demonstrates a similar dedication to Lennie. Even though Lennie makes George frustrated and irritated, George remains loyal. He tells Lennie, “When I think of the swell time I could have without you, I go nuts. I never get no peace.” Yet George doesn’t abandon Lennie. He sacrifices his potential good times out of loyalty for Lennie.

Another similarity is responsibility. Candy feels responsible for his dog’s well-being. When the other workers complain that the dog smells terrible and can barely walk, Candy lets someone else kill him. This decision weighs heavy on Candy's mind. After it happened, Candy admits, “I ought to of shot that dog myself, George. I shouldn’t ought to of let no stranger shoot my dog.”

By the end of the story, George faces a similar dilemma. Feeling responsible for Lennie’s well-being, George doesn’t make the same mistake as Candy. George shoots Lennie himself. He doesn't allow a “stranger”—or Curley—to do it.

In general, the respective relationships have a comparable power dynamic, with Candy assuming power over his dog and George exercising power over Lennie.