Go look at a fan site, there are links to them under the community news tab to the left. These sites have all kinds of information. Such as which pets can, possibly, learn to cast spells on their own. Another thing, some pets add a card, or have skills that add cards to your spell deck; if you cast one of these cards the game is set up to have the pet look like it is casting the spell.
As for pets casting spells on their own, it is horribly horribly unreliable. It is random! My pets have cast sprightly after I have healed, cast a blade after I attacked, and cast two traps on the same monster (I was shocked that it actually cast two spells in the same battle). I would much rather the pet add spell cards to my deck, or just boost my stats like an extra piece of equipment.
Kind of reminds me of that other one-liner post in the PvP section. Well, riddle me this: If every pet had every power, what might happen? Answer: Doomsday for the Arena. Everyone would have manic Spritely powers, completely devastating damage boosts, insane Resist, to the point of about 5,000,000 players quitting from frustration. The death of wizard101.
For want of a stat, the Arena was lost. For want of the Arena, the Wizards were lost. For want of the Wizards the money was lost. For want of the money the Spiral was lost. And all for the want a single pet stat. Sorry, but the game won't always go your own way. If it was a card that came with the pet, then it will show up in your spell deck. If it was a may-cast talent, then max out his Strength, Agility, Speed, etc. That should make him cast it more. Or, you can find the trigger spell. Chances are, when you cast it, he will cast it.