Place the oven rack in the upper third of the oven. Turn on the broiler to high. Line a baking sheet with parchment paper.
Spoon 1 heaping tablespoon of the artichoke-crab mixture onto each baguette slice. Broil until the cheese begins to melt, about 3 minutes. Watch very carefully the entire time so that the toasts do not burn. Sprinkle with remaining 1 tablespoon fresh parsley. Serve warm or at room temperature.
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