To unlock all farming slots in Monster Hunter World, you need to complete 2 optional quests. The first one is called Persistent Pests and is a 4 Star quest. By completing it you will be able to farm two plants, mushrooms or other things at the same time.
To unlock the third farm slot you need to progress further in the story until you get to 7 Star optional quests. Complete Talons of Ire and Ice and report back to the Botanical Research Center. You can now grow and farm in three slots at the same time.