"Clear Monster Park 3 Times"

What exactly does "clear monster park" mean, for Arnello's requirement?

I took 3 characters on 3 runs today. Levels 210 (party leader), 201, and 200. We went in the 200-209 dungeon.
The 210 did all the fighting, the other two just tagged along (but were in the same map as the attacker).
All got the "Clear" and "you've cleared all stages, exit through the portal" message when the boss map was cleared.
All went into the bonus stage (automatically) on the one run where the rhino appeared.
But only the level 210 completed Arnello's quest.

Why didn't the others' clears count?
Are they supposed to use another dungeon?
Do they have to attack (the boss? all enemies? some percentage of enemies?)
Do they have to lead the party?

Also, is there any way to know whether your Clear is being counted, before wasting three runs? When entering it always says "you have cleared Monster Park 0 times today", even for the 210 who did have them count. And the quest helper doesn't even have a counter.

Maybe you need to solo the thing? Since only the Party Leader got the Clears.

So, apparently, just one run didn't count.
I took the other chars for solo runs, and one (each) was enough to complete the quest.

I am guessing that the run that didn't count for them was the one where the rhino appeared. When the party leader went out the final portal, everyone was transported to the bonus map. So, the other two characters didn't pass through that portal, and therefore that run didn't count for them.

Still wish there was a way to check how many times one has cleared the Monster Park. If I'd known just one run didn't count I'd have taken all three chars through again. Would have been faster and easier.