Whisk together the vinegar, mustard, garlic, red pepper flakes, thyme, and parsley in a medium bowl. Slowly whisk in the olive oil until emulsified. Season with salt and pepper. The marinade can be made 4 hours in advance and chilled.
Brush both sides of the mushrooms with the oil and season with salt and pepper. Place the mushrooms on the grill, cap side down, and grill until golden brown and slightly charred, 4 - 5 minutes. Turn the mushrooms over and continue grilling until just cooked through, 3 - 4 minutes longer.
Remove the mushrooms from the grill and cut into 1/2" thick slices. Place the mushrooms in a bowl, add the marinade and toss to coat.
Let marinate at room temperature for 30 minutes before serving. The marinated mushrooms can be made 4 hours in advance and chilled.