Is it possible to pick a vehicle as your commander?
Nope, it can't be used as a commander. And for the record, most planeswalker can't be used either; only those who explicitly state that they can. It's stated that way in the Comprehensive Rules:


  903.3. Each deck has a legendary creature card designated as its commander. This designation is not a characteristic of the object represented by the card; rather, it is an attribute of the card itself. The card retains this designation even when it changes zones.
  Example: A commander that’s been turned face down (due to Ixidron’s effect, for example) is still a commander. A commander that’s copying another card (due to Cytoshape’s effect, for example) is still a commander. A permanent that’s copying a commander (such as a Body Double, for example, copying a commander in a player’s graveyard) is not a commander.
  
  903.3a Some planeswalker cards have an ability that states the card can be your commander. This ability modifies the rules for deck construction, and it functions before the game begins. See also rule 113.6m.


Of course, Commander is a casual format, so if your playgroup likes your idea of a Parhelion II Commander deck, they might allow it as a house rule.