The Sheep and the Goats
One thing that this teaching has in common with many of Jesus' parables is the binary between the "acceptable" (sheep) and the "unacceptable" (goats) -- see, as examples, Two Prayers  or   The Weeds  or  The Net . But binaries scare us. We are much more comfortable with grey than we are with black and white. I am sure that things were no different in Jesus' day. 

And if we take Jesus' words at face value, we are certain to become even more uncomfortable. Neither the sheep nor the goats were anticipating the grounds for the judgment that they received -- good or bad. Every person reacts with surprise in the final judgment! Perhaps that's one of the reasons for Jesus' famous warning to avoid   judging one another ! It certainly is in keeping with the theme of surprise-in-judgment that shows up elsewhere in Jesus' words ( The Bridesmaids , The Banquet or The Laborers are examples along with this teaching ).

In a culture that encourages us to be self-sufficient and to hide our needs from others, even the direct application of this teaching will almost certainly involve all manner of surprises.

​​But also notice that Jesus doesn't give any of his trademark parable-indicators here. As a result, it might be reasonable to conclude that this isn't actually a parable at all! Jesus might simply be giving it to us straight concerning the final judgment! (scary thought!)
Matthew 25:31 (NIV) “When the Son of Man comes in his glory, and all the angels with him, he will sit on his glorious throne. 32 All the nations will be gathered before him, and he will separate the people one from another as a shepherd separates the sheep from the goats. 33 He will put the sheep on his right and the goats on his left.
34 “Then the King will say to those on his right, ‘Come, you who are blessed by my Father; take your inheritance, the kingdom prepared for you since the creation of the world. 35 For I was hungry and you gave me something to eat, I was thirsty and you gave me something to drink, I was a stranger and you invited me in, 36 I needed clothes and you clothed me, I was sick and you looked after me, I was in prison and you came to visit me.’
37 “Then the righteous will answer him, ‘Lord, when did we see you hungry and feed you, or thirsty and give you something to drink? 38 When did we see you a stranger and invite you in, or needing clothes and clothe you? 39 When did we see you sick or in prison and go to visit you?’
40 “The King will reply, ‘Truly I tell you, whatever you did for one of the least of these brothers and sisters of mine, you did for me.'

41 Then he will say to those on his left, ‘Depart from me, you who are cursed, into the eternal fire prepared for the devil and his angels. 42 For I was hungry and you gave me nothing to eat, I was thirsty and you gave me nothing to drink, 43 I was a stranger and you did not invite me in, I needed clothes and you did not clothe me, I was sick and in prison and you did not look after me.’
44 “They also will answer, ‘Lord, when did we see you hungry or thirsty or a stranger or needing clothes or sick or in prison, and did not help you?’
45 “He will reply, ‘Truly I tell you, whatever you did not do for one of the least of these, you did not do for me.’
46 “Then they will go away to eternal punishment, but the righteous to eternal life.”