With two breeding pairs, you could breed each pair and then breed the offspring of each pair together. Or breed the offspring from one pairing to one of the adult birds from the other pairing, since they are unrelated. You might even stretch it out one more breeding cycle by breeding back to the other side of the unrelated pair. After that, you should add new blood, because all your birds are too closely related.
Example of possible unique breeding pairs:
Pair 1 (Male A and Female B) -> Offspring AB
Pair 2 (Male C and Female D) -> Offspring CD
Pair 3 (Male AB and Female CD) -> Offspring ABCD
Pair 4 (Male A and Female CD) -> Offspring ACD
Pair 5 (Male C and Female AB) -> Offspring ABC
Pair 6 (Male AB and Female D) -> Offspring ABD
Pair 7 (Male CD and Female B) -> Offspring BCD
Pair 8 (Male A and Female BCD) -> Offspring ABCD
Pair 9 (Male C and Female ABD) -> Offspring ABCD
Pair 10 (Male ACD and Female B) -> Offspring ABCD
Pair 11 (Male ABC and Female D) -> Offspring ABCD
With just two starting pairs, you can get a decent number of unique pairings before you start to veer into inbreeding territory ... assuming all four birds are completely unrelated.
If you double the number of birds to four pairs, the number of possible combinations also goes way up, since you have even more unrelated breeding lines to work with. However, you must keep really good records and hope to get the right birds from each breeding pair to make this work. In practice, you will likely have significantly fewer viable matches, since you may get no female offspring, no offspring that are of suitable quality to breed, or be unable to keep the offspring from a pair for some other reason. For example, one of your breeding birds might die young or be infertile or show signs of genetic issues that you don't want to pass on. And not all pairings will successfully bond, especially if you are trying to get the same parrot to switch mates multiple times. If you are sticking with one mate for each bird, that would significantly cut down on the possible breeding pairs and mean that you would need new blood much sooner.
Lastly, if you have limited information regarding your birds' true parentage, you might already have closely related birds especially if they were acquired from the same location. This is difficult to account for, other than by bringing in new blood from proven breeding lines.