Not according to the rule you posted (bolding mine)... "no team manager/coach or player will interact with a spectator or parent IN or outside of the dugout during the game."
You're not answering the question. Here's a situation... Jimmy makes a good play to end an inning. As he's running to the dugout a parent (dad, mom, fan, whatever) says "Great play Jimmy!". Jimmy looks up and says "Thanks!". Are you telling me that based on that interaction, the umpire is going to eject Jimmy from the game AND throw the parent out of the facility? This is a yes/no question. I'm not asking whether they CAN, I'm asking if they WILL.
Sigh. Yes, they suspended him for an "altercation with a fan". They did not suspend him for "going into the stands". Semantics? Yes. But that's what this entire back and forth has been about. IN MY OPINION "Going into the stands" does NOT mean "reaching an arm through/around/over a barrier".