The only pet that "grows" is the hatchling, which transforms into a strider you can ride on after you do the strider quest when the hatchling is level 55. -And of course if you are a clan leader of a castle holding clan, you can fly on it as a wyvern. -But you can only mount it from the castle. It cost 25 B crystals.
On the Beast farm near Rune, you can tame the animals that look like the baby pets (I think they're around level 65). When you feed them with what baby spice and crystal spice? They grow, and maybe you succeed in taming one, or it just turns into a multi hp mob that's gonna kill you unless you kill it first.
If you manage to tame it, it may buff you and heal/recharge you depending on whether you're a mystic or a fighter class. Enjoy it while you can, because it might decide it doesn't like you and disappear.