Do plants grow while I'm in another world?

Time in the "base world" actually has to pass for things to grow, corruption/hallow to spread, etc. For anything to happen basically.

So no, plants won't grow if you're off looting someworld else.

An addition/exception by Bob from the comments below: The player doesn't have to be physically in the world; the world just has to be running. One can leave the world running using the dedicated server launcher.

The world is not actually being simulated while you are not in it, so no time is passing and nothing else will actually happen. Even if you run a dedicated multiplayer server, the simulation stops if there are no players in the world. You can imagine it would be problematic if, for example, the corruption continued to spread while you were not playing the game. The same principle applies to plants. They will not grow when the world is not loaded with a player in it.