Place the potato cubes in a large pot and cover with cold water. Bring to a boil over high heat, then reduce to a simmer and cook for 10-12 minutes until tender but not mushy.

While the potatoes cook, hard-boil the eggs: place them in a small saucepan, cover with water, bring to a boil, then remove from heat, cover, and let sit for 10 minutes. Cool under cold water, peel, and chop.

In a medium bowl, toss the cucumber slices with a pinch of salt and let sit for 5 minutes to draw out excess moisture. Rinse and pat dry.

Drain the cooked potatoes and transfer to a large mixing bowl. Mash lightly with a fork or potato masher, leaving some chunks for texture.

Add the chopped eggs, cucumber, grated carrot, and diced ham to the bowl with the potatoes.

In a small bowl, whisk together the mayonnaise, rice vinegar, salt, and pepper until smooth.

Pour the dressing over the potato mixture and gently fold everything together until well combined.

Cover the bowl and refrigerate for at least 30 minutes to allow the flavors to meld.

Before serving, give the salad a quick stir and garnish with fresh parsley if desired.