Scrub potatoes with vegetable brush and cut into 1 1/2" chunks. Place in saucepan and cover with water. Bring to gentle boil over medium high heat. Cook for 10-15 minutes, uncovered, until potatoes can be pierced easily with a fork.
Meanwhile place mustard, vinegar, garlic, parsley, salt, and pepper in medium size mixing bowl. Whisk to combine. Slowly whisk in the olive oil. Taste and adjust seasoning as necessary.
When potatoes are done, remove pan from heat and drain well. I use the lid held at an angle to drain the pot, not a colander.
Put hot potatoes in bowl with dressing and toss gently with a rubber spatula to evenly distribute dressing. The warm potatoes should absorb the dressing and it should have a somewhat creamy consistency.
Top with chives or green onions and serve warm or at room temperature.