No, the Spies are not just the kids who inform on their parents. All kids are supposed to do that. Rather, the Spies are a youth organization run by the Party. It appears that a person becomes a Spy at a young age and then later graduates into the Youth League. So the Spies are supposed to be something like the Hitler Youth -- a group that is meant mainly to indoctrinate the young in the ways of the Party.
You can see hints about this in Chapter 2. We are told that the Spies have been hanging banners for Hate Week. We also see the Parsons' son, who is nine years old, wearing his Spies uniform. This shows us that the Spies are a formal group that do things other than spying on their parents.
Here's a line from Chapter 6 that tells shows the function of the Spies:
By careful early conditioning, by games and cold water, by the rubbish that was dinned into them at school and in the Spies and the Youth League, by lectures, parades, songs, slogans, and martial music, the natural feeling had been driven out of them.