No
And the rule I'm breaking would be...
You aren't an item, you're a creature. You provoke attacks of opportunity for exiting threatened spaces in combat. To pick you up and move you on its action, the elemental would need to grab you, pin you, and then move the grapple... which doesn't change the rules on creatures in combat provoking attacks of opportunity for exiting threatened spaces.
If you climbed aboard on your own action (assuming the DM even allowed you to do so), you'd be engaging in mounted combat... which doesn't change the rules on creatures in combat provoking attacks of opportunity for exiting threatened spaces.
If the elemental had the
Fling Ally feat, he could launch you to where you want to go much more quickly than by winning a grapple with you and moving the grapple. Of course, if you read the feat, you'll notice that it doesn't change the rules on creatures in combat provoking attacks of opportunity for exiting threatened spaces.
You're trying to save actions by making someone else move you, fair enough. Trouble is, that tends to be pretty inefficient to accomplish. Even if you do manage it, unless your mode of move explicitly allows you to break them, you follow the usual movement rules. None of these options explicitly enables you to avoid AoOs, so you're going to be provoking them.
