Bring a large pot of salted water to a boil.
Add the pasta and lower the heat to medium-low to maintain a slight boil. Cook the pasta until it reaches al dente stage.
Drain the pasta, then rinse with cold water several times. Add the frozen peas directly to the cooling pasta so that they thaw as the pasta cools.
Dice the onion into uniform pieces no larger than 1/4 inch.
Then dice the celery larger (1/4 to 1/2 inch pieces). Chop the cheese into 1/2 inch pieces.
Roughly chop the fresh parsley.
Open the cans of tuna and drain them well. Place the tuna in a small bowl and break it up into large flaky pieces.
Place the cooked pasta with peas in a large bowl. Add the red onions, celery, and cheese to the bowl and toss lightly.
Pour the dressing over and gently stir until everything is well coated.
Add in the flaked tuna and chopped parsley, and stir a few more times.