Welcome to the forum and thank you so much for adopting instead of buying a baby!
I am afraid that if your parents do not allow you to lock the dog for a few hours, you can't let him out of its cage. It would be too dangerous. Having said that, I have dogs (too many

) and they are all rescues so it's not as if I trained them from puppyhood not to react to 'prey' animals (I also have cats but they are locked when the birds are out) and they have all learned. The last one is a 15 year old pocket pitbull female that I took in fairly recently and she used to go after both the cats and the birds when she first came (pitbulls have a high prey drive) but she learned not to. She still wants to (you can see it in her body language) but she stops herself from doing it when given a loud SHHHHHH! followed by a "NO! Leave it alone!" as soon as she sits up and tenses and I am sure that in a few more months she will even stop this so you might be able to teach the dog not to react. My golden retriever, whom I've had for many years, learned to freeze when a bird perches on him - and he belongs to a bird hunting breed so it's not impossible. Of course, they are always closely supervised by me when the birds are out (I keep them all in the kitchen by closing the baby gate I have at the entrance and I sit in the kitchen with them).
As to training him... well, I'll be honest with you, I am always of two minds when it comes to training aviary birds (which is what cockatiels are - they are not companion parrots). On the one hand, I do completely understand the human's desire for the closer relationship and ease of handling that implies having a bird step up to a finger (I love birds and I also crave that) and I do firmly believe that taming them so they are not afraid of us is essential for their wellbeing (because of the chronic stress that living with a giant predator alien must mean to them) but, actually, this can be accomplished without any training whatsoever. It just takes patience - lots and lots of patience. I have a few birds that I cannot handle AT ALL but they no longer fear me - quite the contrary, they trust me. Not with the same deep trust that human-imprinted birds have for their chosen humans but they no longer try to bite me or get away from me and they even obey a few commands so they are able to come out of their cages to fly for hours because they will go back into their cages on their own when I tell them to "Go home". I have a parent-raised GCC that learned to go back to her cage when told to and a pair of quakers that I got recently that do not like hands and would not even step up to a stick but I have taught them to step up to my hand (kept flat) covered with 'their' kitchen towel (they have been clipped all their lives and could not fly at all when they first came -they are better now- so, in order for me to put them back on their cage, I had to teach them to step up to something. So, taming and teaching them to trust a human is entirely possible even with parent-raised birds (which is what your tiel is). But, in order for you to be able to do this, you need to be able to let him out of its cage (for some birds, you have to wait weeks and weeks of opening the cage and just wait for them to feel comfortable enough to come out on their own) and, if you cannot do that, there is very little you can do in terms of teaching the bird to step up or anything.
But you can easily teach him to take treats from your hand. All it takes is the right diet and that means no free-feeding protein food because, if you do, for one thing you will end up destroying its liver and kidneys, and, for another, you will never be able to train it. So re-evaluate its diet and find what his high value item is and it will be super easy to get him to take a treat from your hand.
Having said all that, I would like to go a little further on my advice. It deviates from your question but my goal has always been to benefit birds by improving their lives and, for a cockatiel, that means having another bird. Cockatiels are not companion parrots, they are aviary, and even when they are hand-raised and imprinted to a kind human, they are never truly happy without having, at the very least, a mate (best thing for them is a small flock but that is impractical in captivity). It has nothing to do with what the human does or doesn't do - it's the way they evolved and you cannot change evolution. A cockatiel can learn to trust and even love its human, to step up, to perch on its human's shoulder, to enjoy head scratches, etc but what the bird feels for its human will NEVER compare to the enrichment of having a companion of their own species means. So, I would ask you to think a bit about this and see if you can find it in your heart to get him a mate. He is VERY lonely and stressed out as he is now... he has nobody and they need 24/7/365 company. In truth, for a cockatiel to be happy all it needs is a mate, a large flight cage, a good diet and a solar schedule.