Soak the red onion in ice water for 20-30 minutes if you want your onions to taste more mild. Drain the water and add the onion to a serving bowl.
Microwave the peas according to package directions. Drain and rinse the peas with cold water and add them to the bowl with the onions.
Wash and thinly slice the radishes with a mandolin and add them to the bowl with the prepared peas and red onions.
In a clean bowl or glass measuring cup, whisk together the mayonnaise, lemon juice, white vinegar, Parmesan cheese, sugar, dill, salt and pepper to form the dressing. Add the dressing to the pea salad and stir until well combined.
Refrigerate the salad for 2 hours or overnight before serving. Pea salad can be stored in an air-tight container in the refrigerator for up to three days.