Mother Parasect are often found with their baby Paras clinging on to the giant mushroom on their back. The pincers of the babies poke small holes in the soft mushroom, spores come out, and this is how the young Paras get their own mushrooms.
Interestingly enough, a theory is going around challenging the belief that Parasect are entirely controlled by the parasitic mushroom. Some researchers have done research in to the possibility that they are instead codependent on each other.
The biggest supporting factor of the original theory is that if you remove the mushroom off of a Parasect, it will stop moving. The new theory states that, potentially, this is a survival tactic to deter predators from the now defenseless Pokémon and it will stay that way until it comes in to contact with spores and grows a new mushroom.