Cards entering the battlefield simultaneously and "Enter the Battlefield" effects
Your opponent is correct, whenever multiple objects enter the battlefield at the same time, their enters the battlefield triggers will all see all of them as on the battlefield.


  603.6a Enters-the-battlefield abilities trigger when a permanent enters the battlefield. These are written, "When [this object] enters the battlefield, ..." or "Whenever a [type] enters the battlefield, ..." Each time an event puts one or more permanents onto the battlefield, all permanents on the battlefield (including the newcomers) are checked for any enters-the-battlefield triggers that match the event.


And:


  603.3b If multiple abilities have triggered since the last time a player received priority, each player, in APNAP order, puts triggered abilities he or she controls on the stack in any order he or she chooses. (See rule 101.4.) Then the game once again checks for and resolves state-based actions until none are performed, then abilities that triggered during this process go on the stack. This process repeats until no new state-based actions are performed and no abilities trigger. Then the appropriate player gets priority.


Note that the trigger happens AFTER the cards actually enter the battlefield. There is no time that Valakut was on the battlefield but the other Mountains weren't; so by the time Valakut's "enters the battlefield" triggers, they are all on the battlefield. If it did not work this way, any card such as Elvish Visionary that had "when [CARDNAME] enters the battlefield..." wouldn't work, because its ability doesn't technically exist until the card was put on the battlefield.

This is also a well-known combo for Valakut.

This means that this works for Champion of the Parish as well; the Champions would get 6 +1+1 counters in that case.

See similar question here.