I, personally, never recommend getting a parrot with the intention of caging it with a current member of your flock. Parrots have a wide range of personalities, just like us, and even if you were to get another tiel... there's no guarantee that you would ever be able to cage him/her with your current tiels. Parrots are also very delicate... all it would take is one fight while you're away or out of hearing range for you to come back to a few dead or severely injured birds. Even after a long introduction period... cages next to each other, extended playtime together on a daily basis... it's like people, we might get along great but once we move in with each other, all hell can break loose.

In trying to think of another type of parrot that I would even try to recommend to cage together... I'm not thinking of any. There's so many things that would need to match...
- Parrot size (budgies are too small compared to them)
- Beak size (conures are the same size body but their beaks are bigger)
- Temperament (quakers and lovebirds can be extremely territorial)
- Diet (tiels, like budgies, need more seeds in their diet than other birds... also, a fortified pellet mix should be a part of every parrot's diet and tiels usually need a smaller size pellet than conures, quakers, etc. do)
- Bar thickness (my quaker parrot tore bars off of a typical cockatiel-sized cage), bar spacing, and perch diameter requirements

I can't think of anything. If you have a LiveJournal account (or want to make one, it's free), there's a great parrot community there that you could talk to if you'd like to get more opinions. http://community.livejournal.com/parrot_lovers/

And I'd just like to throw a general reminder out there, in case you do get another bird... remember to quarantine!

- Keep the new bird in a completely different air space for 30 days. If you have central air/heat, pick up a few antibacterial air filters and tape them over the air vent in your current tiels' room and your new bird's room. Wash your hands between rooms. Don't hold the new bird against your shirt and then go hold one of your original birds against the same shirt. Parrots are very good at hiding signs of illness, and could be harboring just about anything.
- You should already have the contact information of a certified avian vet, since your current tiels should be seeing one on a yearly basis, but if you don't here's a certified avian vet finder: http://www.aav.org/vet-lookup/ . Take the new bird in for a complete check-up, and don't even introduce him/her to the same room until that and the 30-day quarantine are up.

It is not a matter of species of birds, but temperament and individual birds. More social birds such as lovebirds and parakeets are a good choice. However, you want to be sure that they are not too aggressive and used to being kept within a "flock."

Larger parrots can be risky because if they get aggressive with your smaller cockatiels, they could kill them.

However, allow me to add a cautionary tale. I sold my Indian ringneck (which is about the same size as a cockatiel) to a very nice woman, member of a bird club and had other birds. She knew my bird had not been kept with other birds for several years. Before that, he had his own cage in a pet store.

Somehow, she allowed my bird out with one of her other birds. He attacked and nearly killed it. We ended up having to find another home for my bird. So, adding other birds should always be done with care and a watchful eye.