Bring a 3-quart / 2.8 L saucepan of well-salted water to a boil. Add the potato and carrot cubes and cook for 8 to 10 minutes, until a knife slides in with slight resistance. Check at 8 minutes so they don't go past fork-tender.
Add the frozen peas for the last 2 minutes of cooking, then drain everything into a colander.
Spread the drained vegetables on a sheet pan (33 x 46 cm / 13 x 18 inch) lined with a kitchen towel and let them cool completely, about 15 minutes. This stops the cooking and keeps the cubes firm instead of turning to mush.
Meanwhile, lower the eggs into a 1.5-quart / 1.4 L saucepan of simmering water and cook for 9 minutes. Transfer to an ice bath, then peel and chop once cool.
In a large mixing bowl (3-quart / 2.8 L), whisk the mayonnaise, lemon juice, mustard, salt and pepper until smooth.
Fold the cooled potatoes, carrots and peas into the dressing with a silicone spatula, turning gently to coat without crushing the cubes.
Fold in the chopped gherkins, capers and two-thirds of the chopped egg.
Cover and chill for at least 1 hour, ideally overnight, to let the flavors settle.
Before serving, taste and adjust salt. Transfer to a serving dish, top with the remaining chopped egg, scatter with parsley, and drizzle with olive oil.