Hi kelvin and flock! Welcome to the forum.
A clicker is definately not neccasary, all it is is a noise to say 'you did well', instead of this you can click your tongue if you wish, or, what I do is say 'Good' or 'Good girl'. Whether you are making a click noise or verbally praising them, all it is doing is letting them know they did what you have asked them to do, you are proud of them and they now get a treat.
As to training them alone or together, I dont really have experience in training more than one bird at the same time/ at once- I beleive it would be easier for you if you gave the bird you were training 1 on 1 attention for its training session, put it back in its cage or bird room and then start working with the next one, giving it 1 on 1. This way not only are your birds not quarreling with eachother causing distraction for the whole flock at the time of training, but if the bird is getting your full attention and praises and happiness, it makes it a more encouraging experience for the parort, which makes it more willing for the next session.-
Best wishes with your two Ringnecks and your Senagal parrot and hopefully this helps