Another account with a mage/or mage friend
Another account/friend with two seater flying mount
Since the pet dailies are account wide, you do them on any character on your account.
-Make the appropriate amount of Level 1 characters to park at the trainers/fabled pets you want to fight
-I suggest making all your level 1 toons on a PVE server. Make monks because they have faster movement speed. Remember, your friend can still invite you to party to port you if you are on different servers as long as you are on the same faction.
-Get your mage friend to port all your characters to Vale of the Eternal Blossoms. To speed up this process, have all your level 1 toons already pre-made. When it comes time to porting, you can have 2 WoW clients open. The moment you are ported to vale, log onto your account on on your second client to bypass the 20 seconds logout time.
-After all your toons have been ported, get your friend on his/her 2 seater mount to fly you to all the pet trainers/fabled beasts. (You probably will have 20 toons that need to be flown. might be better to spread this burden among your friends)
By now you should have have all your toons parked. When it comes time to do the dailies, have two WoW clients open. The moment you defeat a trainer, log onto your account on your 2nd client to bypass the 20 second log out time. Since the trainers award bags with bandages, you can easily heal if need be.
For the fabled beasts, you cannot heal because they don't reward bags and u will never have bandages. To fix this problem, make sure you have a different battle team (and one that is a guaranteed win) for every beast.
CAUTION/AND OTHER NOTES:
Greyhoof - Valley of the Four Winds:
You cannot park a toon for Greyhoof in Valley of Eternal Four Winds. There are invisible foxes in every agro radius around him. The corpse run back to Greyhoof is also ridiculously long.
Ka'wi the Gorger - Jade Forest:
There are also mobs all around Ka'wi the gorger. To get around this, you must have a pre selected team and be dead at Ka'wi. The moment you click accept to respawn, spam your button to engage Ka'wi. After the battle, you will die to MOBS for sure. The good thing is, the corpse run is a hop, skip, and a jump. Just leave your character dead till the next time you log back in to kill him.
Trainer Seeker Zusshi in Townlong Steppes:
They finally got rid of that stupid patting mob from this trainer. However, there's still an annoying basilisk that sometimes pats in your agro radius.
When you log out, log out in the lake.
It is possible to run your level 1 toon to:
No-No - Vale of Eternal Blossoms
Ti'un the Wanderer - Townlong Steppes
Kafi - Kun Lai Summit
Each trip should take 5-15 minutes. Head west from shrine. Die. Repsawn at spirit guide. Run as far west as possible. If you die, run back to your corpse (this is mad easier/faster because you get a 310 flying mount in ghost form when you die in vale). A Rinse and repeat always ressing at a spirit guide furthest west. When you get to the wall, hug it and die and you should respawn at the top of it.
I started doing this because I'm sick of wasting an hour of afk travel time every day just to do the pet dailies in pandaria