I bought the Pandemic base set, and In The Lab, as I saw it plays up to six people. (Though I notice now that's for team play).
When I got home, I saw the box says that you need both the base set and On The Brink to play.
Is this correct?
Yes, it's correct, in that to use everything in the box, you absolutely need the base set and On the Brink. There are things in there that directly add on to things from On the Brink, and would be useless without it.
But you can still use most of it without On the Brink. You can add in the roles and events, and you can play the lab, solo, and team variants. This is all summarized in the "overview" section at the beginning of the rules, and the other sections make it explicit:
Lab Challenge: "This challenge can be played with just the base game or combined with other challenges, except the Team game."
Solo Game: "The solo game pits you, with a little help from the CDC, against all the diseases. It can be combined with other challenges, except the Bio-Terrorist challenge."
Team Game: "The Team game can be combined with other challenges, except the Bio-Terrorist or Lab challenges."