Candy's relationship with his dog is like George and his relationship with Lennie. What are some similarities besides letting the dog and Lennie...

When George and Lennie walk by the pool in the first chapter, George leads and Lennie follows. When Candy and his dog walk into the bunkhouse in Chapter 3, Candy leads and the dog follows. This simple illustrations shows how the dog is dependent on, and a follower of, his owner. Lennie is also dependent upon George and he will follow George (and obey his commands) wherever he goes. 

The first man was small and quick, dark of face, with restless eyes and sharp, strong features. Every part of him was defined: small, strong hands, slender arms, a thin and bony nose. Behind him walked his opposite, a huge man, shapeless of face, with large, pale eyes, and wide, sloping shoulders; and he walked heavily, dragging his feet a little, the way a bear drags his paws. (Chapter 1) 

Lennie is described as having "paws." This is not the only time Lennie is described in terms of animal characteristics. This shows a clear parallel between himself and the dog. 

Candy tries to protect his dog from the other men (namely Carlson) who wants to shoot the dog, thus putting the dog out of its (and their) misery. George must always protect Lennie as well and this is particularly the case in the end when he is protecting him from Curley. 

Both Candy and George must protect their followers (dog and Lennie) respectively. And they both literally speak for their followers. Note in the second Chapter when George speaks for Lennie in order to avoid any social conflicts. 

The boss pointed a playful finger at Lennie. “He ain’t much of a talker, is he?” “No, he ain’t, but he’s sure a hell of a good worker. Strong as a bull.” Lennie smiled to himself. “Strong as a bull,” he repeated. George scowled at him, and Lennie dropped his head in shame at having forgotten. 

Lennie drops his head the way a dog would after being scolded by its master. 

When Candy first considers letting Carlson kill the dog, he looks to Slim for a reason not to. Slim doesn't have one. Likewise, at the end of the story, Slim acknowledges that George had to kill Lennie. Slim silently and then verbally gives his approval for each mercy killing: 

Candy looked a long time at Slim to try to find some reversal. And Slim gave him none. At last Candy said softly and hopelessly, “Awright—take ‘im.” (Chapter 3) 

Slim said, “You hadda, George. I swear you hadda. Come on with me.” He led George into the entrance of the trail and up toward the highway. (Chapter 6)