Agricola - After/Also rule
No,

The first part is mandatory, the second optional.

The Rules for Family growth (page 5) state:


  Players must have room for offspring in their home before they can
  use this action  – that is, they must have more rooms in their home than they have Family members. After taking Family growth, the player may choose to purchase a Minor Improvement 


Note that in general, a player must be able to use an action to be allowed to place a worker on it. (page 3), 


  A Family member may never occupy an Action space without
  performing its action.


Therefore you cannot place a worker there just to block another player.