The biggest use I could see would be combining Elemental weather immunity with either "Call Darkness" or "Scorched Earth".
For Call Darkness: "A shadow falls across the battlefield, dealing 15 Humanoid damage and turning the weather to Darkness for 9 rounds. During Darkness, all pets are considered Blinded and all healing received is reduced by (an amount)"
So for Call Darkness, we have a battlefield reduction to Accuracy and Healing as a weather effect. Unfortunately, no elementals currently have the ability, so it would require a switch for the benefit offensively. Defensively, if there is a team built on Darkness/Blindness (I think some stuns get boosted to guaranteed, like on the Pandarian Monk pet) an elemental might be enough to break the team comp.
For Scorched Earth: "Scorches the ground, dealing 15 Dragonkin damage and dealing 4 Dragonkin damage to all active pets each round for 9 rounds. During Scorched Earth, all pets count as Burning."
I don't think this combo is quite as strong offensively, which is probably why you find some elemental pets with scorched earth, as it isn't as game breaking. Now, there are some abilities that cause extra damage based on your opponent being burning, like Conflagrate. Defensively, your elemental pet will not be burning, which gives them extra utility against teams based around Scorched Earth and the benefits that it grants.
Defensively, you can counter these abilities without an elemental, by using another weather effect. With some of the powerful combos based around Moonfire and its weather effect, I think I will be using Magic pets as opposed to an elemental unless I'm building a team around one of these two moves.