I would say it depends.
When it comes to tiels and parakeets, my answer is a definite no when it comes to synthetic vitamins.
These birds are adapted to live almost exclusively on fresh, young seeds and sprouts, which means they have adapted to a diet pretty low on vitmins, as opposed to many other birds.
Especially for tiels it seems as like they are very sensitive to vitamin A. Studies have shown that they do better on a diet completely lacking in vitamin A than on one with too much. Too much, in this case, being way less than what many other species require. Too much results in excessive screaming and many other problems. The effects of too much viamin A can be s real pain, both in humans and other animals. (Here's the abstract for one of many studies:
http://jn.nutrition.org/content/133/6/1898.short)
Note that this only goes for the "finished" vitamin A that is mostly found in supplements.
Beta-Carotene, the pre-stage for vitamin A that is found in orange, red and yellow veggies, is pretty much not possible to overdose since the body can easily get rid of any excess.
Also, most pelleted diets contain way too much protein for parakeets and tiels. Their diet should consist mostly of a lot of dirrent kinds of seeds which have been put in water 1-2 days before fed, (change the water and rinse them every 6 hours or so, keep them cool and dark.) as well as leafy greens, herbs, flowers etc.
Too much vitamin A is actually a common problem in other birds as well, and, just as in teals, can lead to a number of behvaioural issues, as well as other stuff you'd rather avoid.
I've actually looked it up, and most pellets contai up to 15-17IU of vitamin A, which is A LOT!
Harissons for example contains only 6,3 IU. With that said, if you are going to feed pellets at all, make sure it is a high quality pellet, and watch out for the vitamin A levels. Complement with veggies rich in beta-carotene.
If you have a bird that's a really picky eater, i might recommend (a good quality) pellets make up a larger part of the diet. If however your bird eats anything i strongly recommend not giving more than about 10%.
Also, be very, very careful with vitamin supplements, and never give supplements these if your bird has pellets in it's diet.

/Steph