OK...since this is a tiel, and will be reluctant to be orally fed or given anything you must also try and keep the head held as firmly as possible. I am left-handed, so the illus are for a lefty. A righty would dose on the birds right side as Prince pointed out, but the hand hold would be similar.
The first 2 pix's are the handhold I use for handfeeding a cockatiel. The same grip would be used for a reluctant tiel as to medicating.
The last 2 pixs are of a mousebird, but it shows the formula being dispenced down the birds right side of the esophagus. I crop/espophagus feed my mousebird chicks because after aspirating some in the past it is the saftest way to dispence fluids. You might want to ask your vet if he has a dispencing tip for oral meds, if needed. The place where I get these tips also sells to vets, and the website is listed on the illustration.
Usually when a syringe tip is touching the inside of the mouth the bird will automatically close the glottis (not sure of the spelling) to keep fluids from going down the wrong way.
Tiels are not fond of juices or sweet things, so do not mix the meds with anything sweet. This will be rejected as strongly as the taste of the meds.
If in the event a bird does get aspirated it can gasp (from lack of air) die quickly. Watch carefully after you give the meds. If the bird is gasping and staggerring you might need to clear the airway. What I have done as soon as I realized this was to grash the bird and put my entire mouth over it's beak and cer and gently blow a quick puff of air into the bird. 95% of the time it clears the airway so that it can breathe. NOt too hard, just a gentle puff or two.