The strongest parallel in Of Mice and Men is between Candy and his old dog.  In fact Candy worries about being done away with if he can no longer be useful.  When he asks to come in with Lennie and George on their plan to have a place of their own, he confides to George, 

"They'll can me purty soon.  Jus' as soon as I can't swamp out no bunk houses they'll put me on the county.

Then he tells George,

"I ought to of shot that dog myself, George.  I shouldn't ought to of let no stranger shoot my dog."

Candy's words hint at the parallel that also exists between Lennie and Candy's dog.  Having been described as having paws and walking like a bear in Chapter One, Lennie's character parallels that of Candy's dog at the end.  For, it is indeed a mercy killing that George performs by shooting Lennie; unlike Candy, however, George does have the courage to shoot his old friend to keep him from suffering while Candy let the cruel Carlson do it. 

In John Steinbeck's Of Mice and Men, when I think of parallelisms in the story, the first thing that catches my notice is the similarity between George and Lennie and a number of the men in the bunkhouse. These men represent the thousands of disenfranchised individuals and families who lost everything in the Great Depression and moved across the face of America to find work and a better life.

Lennie, being very large and very strong, with the intellect of a small child, is so fond of soft things that he ends up destroying them because he cares for them so much. We see this with the dead mouse he finds, which is now "broken" after Lennie "pets" it, with the puppy he kills by mistake, and even with Curley's wife at the end. Lennie understands certain basic things, but he does not understand the extent of his strength, even after breaking Curley's hand.

At the end, when Lennie has accidentally killed Curley's wife, George's love for Lennie becomes evident. He understands that if the other men catch Lennie, especially Curley who already has it out for Lennie, that his friend will meet a terrible end.

Here, in one parallel, George is reminded of the putting down of Candy's dog.

Of the dog, Slim says:

That dog ain't no good to himself. I wisht somebody'd shoot me if I got old an' a cripple...

Lennie and the dog are very similar. They have been attached to someone for many years: Lennie with George, and the dog with Candy. Both can no longer function in the world of which they are a part. They die in the same way: shot in the same place with the same gun.

But Candy had let someone else, a stranger, end his companion’s life and he regretted it, and George is determined not to make that same mistake.  (eNotes.summary&analysis)

And so George, in as loving a way as Lennie was with the mouse and the puppy, decides he must "take care" of Lennie. In the spot next to the water, where they book began, George begins to tell the same tale they have spoken of, time after time, of the house and the farm they will have. George invites the other man to look over the water and imagine that he can see that place; and then he takes the pistol from his pocket and shoots Lennie.

With great difficulty George fires the gun at the place where Carlson had told Candy to shoot the dog, the spot at which the creature would die feeling no pain. George pulls the trigger only after taking Lennie to their dream farm one last time. (eNotes.summary&analysis)

When the others arrive, George lies and says that Lennie had the gun and George took it to defend himself.

In terms of the story, the example of parallelism that strikes me most strongly, is that of Lennie relationship with is animals, and George's behavior toward Lennie at the story's end.

The humanity or mercy in George's actions are shocking and devastating to the reader, that this "innocent" man (Lennie), through no fault of his own, could not survive in the society of people as he harms or kills wherever he goes. But we can also strongly sympathize with George who has given up a great deal to take care of Lennie. Though he complains about him, it is evident that George cares deeply for this unusual man.