Can an Adaptive Automaton trigger Champion of the Parish's ability?
Adaptive Automaton will enter the battlefield already possessing the chosen creature type, which will trigger Champion of the Parish.

Automaton's first ability is a "replacement effect". Relevant rules are under the heading "Replacement Effects". Relevant excerpts:


  614.1. Some continuous effects are replacement effects. Like prevention effects (see rule 615),
     replacement effects apply continuously as events happen—they aren’t locked in ahead of time. Such
     effects watch for a particular event that would happen and completely or partially replace that event
     with a different event. They act like “shields” around whatever they’re affecting.
  
  [...]
  
  614.1c Effects that read “[This permanent] enters the battlefield with . . . ,” “As [this permanent]
     enters the battlefield . . . ,” or “[This permanent] enters the battlefield as . . . ” are replacement
     effects.
  
  [...]
  
  614.6. If an event is replaced, it never happens. A modified event occurs instead, which may in turn
     trigger abilities. Note that the modified event may contain instructions that can’t be carried out, in
     which case the impossible instruction is simply ignored.


For more information about how this interacts with Champion of the Parish, let's look at "Handling Triggered Abilities" -- specifically:


  603.6. Trigger events that involve objects changing zones are called “zone-change triggers.” Many
     abilities with zone-change triggers attempt to do something to that object after it changes zones. During resolution, these abilities look for the object in the zone that it moved to. [...]
  
  [...]
  
  603.6b Continuous effects that modify characteristics of a permanent do so the moment the
     permanent is on the battlefield (and not before then). The permanent is never on the battlefield
     with its unmodified characteristics. Continuous effects don’t apply before the permanent is on
     the battlefield, however (see rule 603.6e).
  
  Example: If an effect reads “All lands are creatures” and a land card is played, the
  effect makes the land card into a creature the moment it enters the battlefield, so it
  would trigger abilities that trigger when a creature enters the battlefield. Conversely, if
  an effect reads “All creatures lose all abilities” and a creature card with an enters-the-
  battlefield triggered ability enters the battlefield, that effect will cause it to lose its
  abilities the moment it enters the battlefield, so the enters-the-battlefield ability won’t
  trigger.


So, in essence, "As Adaptive Automaton enters the battlefield, choose a creature type"  replaces the normal "~ enters the battlefield" action that you perform when summoning a creature with a subtly different action: "~ enters the battlefield and its controller chooses a creature type".

"Adaptive Automaton is the chosen type in addition to its other types" is a static ability just like shroud or flying, so your Automaton is an "Artifact Creature - Human Construct" (or "Construct Human", I suppose, the rules aren't picky) when it enters the battlefield, and this is the type that all triggered abilities (like Champion's) will see.

If you'd like to see an official example of Adaptive Automaton interacting with a Human-enters-the-battlefield ability, look at the Rulings section for Dearly Departed.