Well, the only way I can see the way to do that is this:
Download TwistedMexi's mod that allows all the cheats that were disabled in Seasons patch, then enter this cheat:
sims.get_sim_id_by_name (I have found it working when the pet doesn't have a last name)
You'll get the pet's simID. Turn the aging on for a moment and enter this cheat:
sims.age_up
Elder pets should then die by old age while younger pets age up.
There is a pet age up treat that can be bought in a vet clinic, but it cannot be given to elder pets.
So I really don't know any mod-free alternatives. We can ask the developers to unlock/add a cheat that ages up sims and pets.
